| L(s) = 1 | + 3-s − 2·5-s − 2·7-s − 9-s − 11-s − 5·13-s − 2·15-s − 5·17-s + 6·19-s − 2·21-s − 2·23-s + 3·25-s − 29-s − 33-s + 4·35-s − 12·37-s − 5·39-s + 2·41-s − 10·43-s + 2·45-s − 5·47-s + 3·49-s − 5·51-s + 2·53-s + 2·55-s + 6·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s − 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.516·15-s − 1.21·17-s + 1.37·19-s − 0.436·21-s − 0.417·23-s + 3/5·25-s − 0.185·29-s − 0.174·33-s + 0.676·35-s − 1.97·37-s − 0.800·39-s + 0.312·41-s − 1.52·43-s + 0.298·45-s − 0.729·47-s + 3/7·49-s − 0.700·51-s + 0.274·53-s + 0.269·55-s + 0.794·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.794989892081942223863041814437, −8.341368428246945878026157408956, −8.236105514729567537814408461202, −7.67912836021196888328379699789, −7.20349257718727825632571128986, −7.08763181708918741390224908353, −6.65809159847293973634747588971, −6.23735582669023205209687307485, −5.48921128149683636103455954920, −5.28761535633031239136170163291, −4.80713802539629794889699247792, −4.41748504546212454389317519090, −3.69120549810522477168483411390, −3.56706639613450681825029759454, −2.89569295992795399015189539356, −2.66563600005863088610675817451, −2.06494243503967235910949717449, −1.29277150774566829327766263821, 0, 0,
1.29277150774566829327766263821, 2.06494243503967235910949717449, 2.66563600005863088610675817451, 2.89569295992795399015189539356, 3.56706639613450681825029759454, 3.69120549810522477168483411390, 4.41748504546212454389317519090, 4.80713802539629794889699247792, 5.28761535633031239136170163291, 5.48921128149683636103455954920, 6.23735582669023205209687307485, 6.65809159847293973634747588971, 7.08763181708918741390224908353, 7.20349257718727825632571128986, 7.67912836021196888328379699789, 8.236105514729567537814408461202, 8.341368428246945878026157408956, 8.794989892081942223863041814437
