| L(s) = 1 | − 2·2-s − 7·3-s + 5·5-s + 14·6-s + 8·8-s + 27·9-s − 10·10-s + 33·11-s − 86·13-s − 35·15-s − 16·16-s − 111·17-s − 54·18-s + 70·19-s − 66·22-s − 42·23-s − 56·24-s + 172·26-s − 224·27-s − 450·29-s + 70·30-s + 88·31-s − 231·33-s + 222·34-s + 34·37-s − 140·38-s + 602·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·3-s + 0.447·5-s + 0.952·6-s + 0.353·8-s + 9-s − 0.316·10-s + 0.904·11-s − 1.83·13-s − 0.602·15-s − 1/4·16-s − 1.58·17-s − 0.707·18-s + 0.845·19-s − 0.639·22-s − 0.380·23-s − 0.476·24-s + 1.29·26-s − 1.59·27-s − 2.88·29-s + 0.426·30-s + 0.509·31-s − 1.21·33-s + 1.11·34-s + 0.151·37-s − 0.597·38-s + 2.47·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.03505674980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03505674980\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 7 T + 22 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 p T - 2 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 43 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 111 T + 7408 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 70 T - 1959 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 42 T - 10403 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 225 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 88 T - 22047 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T - 49497 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 178 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 411 T + 65098 T^{2} + 411 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 708 T + 352387 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 480 T + 25021 T^{2} + 480 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 812 T + 432363 T^{2} + 812 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 596 T + 54453 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 358 T - 260853 T^{2} - 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 425 T - 312414 T^{2} + 425 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 972 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 960 T + 216631 T^{2} + 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 709 T + p^{3} T^{2} )^{2} \) |
| 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
−10.97597802659204349453793429701, −10.28291027096125631015614238412, −9.708730508944235052479769476908, −9.517582289085301066314151749700, −9.226340764849911306304120499455, −8.813106439532375879681206356980, −7.71340178495654491594556280227, −7.65101213372093686521049118286, −7.24343891583408931301612625709, −6.42805654503691880915049169995, −6.34263661275877719350159410579, −5.45216355471158540118930984995, −5.39797891837241883603777503697, −4.52796991343914355708942470553, −4.26549452732571677715573404262, −3.49332780714693980358293537844, −2.34788700223382175297150966520, −1.97301140764866275216442502702, −1.10848444636484189351437831143, −0.080621967312098126723051860574,
0.080621967312098126723051860574, 1.10848444636484189351437831143, 1.97301140764866275216442502702, 2.34788700223382175297150966520, 3.49332780714693980358293537844, 4.26549452732571677715573404262, 4.52796991343914355708942470553, 5.39797891837241883603777503697, 5.45216355471158540118930984995, 6.34263661275877719350159410579, 6.42805654503691880915049169995, 7.24343891583408931301612625709, 7.65101213372093686521049118286, 7.71340178495654491594556280227, 8.813106439532375879681206356980, 9.226340764849911306304120499455, 9.517582289085301066314151749700, 9.708730508944235052479769476908, 10.28291027096125631015614238412, 10.97597802659204349453793429701
