| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 6·7-s − 4·8-s + 2·9-s − 4·10-s + 2·11-s + 6·12-s + 12·14-s + 4·15-s + 5·16-s + 2·17-s − 4·18-s + 6·20-s − 12·21-s − 4·22-s + 10·23-s − 8·24-s − 25-s + 6·27-s − 18·28-s + 14·29-s − 8·30-s + 2·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 2.26·7-s − 1.41·8-s + 2/3·9-s − 1.26·10-s + 0.603·11-s + 1.73·12-s + 3.20·14-s + 1.03·15-s + 5/4·16-s + 0.485·17-s − 0.942·18-s + 1.34·20-s − 2.61·21-s − 0.852·22-s + 2.08·23-s − 1.63·24-s − 1/5·25-s + 1.15·27-s − 3.40·28-s + 2.59·29-s − 1.46·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9480121494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9480121494\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.76841685050946347739577926394, −12.74288416180616246752317165507, −12.05124367402013418816721060918, −11.53010079939354615506980145165, −10.41330346868324433677034993930, −10.30373049374836459550272032171, −10.02133221792227399193976550963, −9.288651859170860099312623928862, −9.087533774643692909826836873119, −8.714363949709060854494833972739, −8.153798080592500324190851358899, −7.24255812838083619280230448074, −6.72852204899106102727910528364, −6.57173697255431825899780122578, −5.88077663238468884333050191701, −4.86706772317437258134043427856, −3.51790198156131782015181594839, −3.02499588644589600905987236169, −2.58463664147073789040992382391, −1.19921881510124869737098059482,
1.19921881510124869737098059482, 2.58463664147073789040992382391, 3.02499588644589600905987236169, 3.51790198156131782015181594839, 4.86706772317437258134043427856, 5.88077663238468884333050191701, 6.57173697255431825899780122578, 6.72852204899106102727910528364, 7.24255812838083619280230448074, 8.153798080592500324190851358899, 8.714363949709060854494833972739, 9.087533774643692909826836873119, 9.288651859170860099312623928862, 10.02133221792227399193976550963, 10.30373049374836459550272032171, 10.41330346868324433677034993930, 11.53010079939354615506980145165, 12.05124367402013418816721060918, 12.74288416180616246752317165507, 12.76841685050946347739577926394
