| L(s) = 1 | − 2·3-s + 3·7-s − 7·9-s + 11-s + 4·13-s − 5·17-s − 14·19-s − 6·21-s + 7·23-s + 20·27-s − 12·29-s − 2·33-s + 4·37-s − 8·39-s − 7·41-s + 43-s − 9·47-s + 12·49-s + 10·51-s + 16·53-s + 28·57-s + 23·59-s + 25·61-s − 21·63-s − 9·67-s − 14·69-s − 7·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.13·7-s − 7/3·9-s + 0.301·11-s + 1.10·13-s − 1.21·17-s − 3.21·19-s − 1.30·21-s + 1.45·23-s + 3.84·27-s − 2.22·29-s − 0.348·33-s + 0.657·37-s − 1.28·39-s − 1.09·41-s + 0.152·43-s − 1.31·47-s + 12/7·49-s + 1.40·51-s + 2.19·53-s + 3.70·57-s + 2.99·59-s + 3.20·61-s − 2.64·63-s − 1.09·67-s − 1.68·69-s − 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 3 T - 3 T^{2} - 5 T^{3} + 96 T^{4} - 5 p T^{5} - 3 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 - T + 5 T^{2} + p T^{3} + 4 p T^{4} + p^{2} T^{5} + 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + p T^{2} - 10 T^{3} + 116 T^{4} - 10 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 5 T + 38 T^{2} + 140 T^{3} + 809 T^{4} + 140 p T^{5} + 38 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 14 T + 97 T^{2} + 482 T^{3} + 2135 T^{4} + 482 p T^{5} + 97 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 T + 61 T^{2} - 291 T^{3} + 1604 T^{4} - 291 p T^{5} + 61 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 89 T^{2} - 138 T^{3} + 3524 T^{4} - 138 p T^{5} + 89 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 7 T + 3 p T^{2} + 729 T^{3} + 6900 T^{4} + 729 p T^{5} + 3 p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 63 T^{2} + 15 T^{3} + 2116 T^{4} + 15 p T^{5} + 63 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 159 T^{2} + 993 T^{3} + 10384 T^{4} + 993 p T^{5} + 159 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 23 T + 400 T^{2} - 4428 T^{3} + 40409 T^{4} - 4428 p T^{5} + 400 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 25 T + 379 T^{2} - 4015 T^{3} + 34616 T^{4} - 4015 p T^{5} + 379 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 9 T + 234 T^{2} + 1728 T^{3} + 22559 T^{4} + 1728 p T^{5} + 234 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 7 T + 3 p T^{2} + 819 T^{3} + 18840 T^{4} + 819 p T^{5} + 3 p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 161 T^{2} + 666 T^{3} + 13379 T^{4} + 666 p T^{5} + 161 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 277 T^{2} + 802 T^{3} + 31400 T^{4} + 802 p T^{5} + 277 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 307 T^{2} + 3058 T^{3} + 35595 T^{4} + 3058 p T^{5} + 307 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 459 T^{2} + 5798 T^{3} + 64539 T^{4} + 5798 p T^{5} + 459 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.65820857910524980839070467980, −5.59015791879992188041981626276, −5.47201669515894411186366257091, −5.33192571880601818988694783438, −5.14313241598963574828292593702, −4.77099426953307142103750450811, −4.61118468742216119235109302409, −4.56530871161344651718902095636, −4.23757955026529955856155629988, −4.14237666358822470530569047177, −3.88399987889765925427574582877, −3.80904450176876793520947410267, −3.66184938774682245478658945921, −3.23726575552015637417312952341, −3.19694734818075751984747298436, −2.78328511275617759694178860387, −2.61816352023936586116364193759, −2.41840197817906227223670620218, −2.15726039490590524575136680921, −2.15119640099294174958029969636, −2.13725468986192389147395190118, −1.34938684344712176569677399158, −1.28329529387591255234836991924, −1.11936970966046570984219794831, −0.903041997917561280878245568300, 0, 0, 0, 0,
0.903041997917561280878245568300, 1.11936970966046570984219794831, 1.28329529387591255234836991924, 1.34938684344712176569677399158, 2.13725468986192389147395190118, 2.15119640099294174958029969636, 2.15726039490590524575136680921, 2.41840197817906227223670620218, 2.61816352023936586116364193759, 2.78328511275617759694178860387, 3.19694734818075751984747298436, 3.23726575552015637417312952341, 3.66184938774682245478658945921, 3.80904450176876793520947410267, 3.88399987889765925427574582877, 4.14237666358822470530569047177, 4.23757955026529955856155629988, 4.56530871161344651718902095636, 4.61118468742216119235109302409, 4.77099426953307142103750450811, 5.14313241598963574828292593702, 5.33192571880601818988694783438, 5.47201669515894411186366257091, 5.59015791879992188041981626276, 5.65820857910524980839070467980
