L(s) = 1 | − 2·2-s + 3·3-s + 3·4-s − 6·6-s + 5·7-s − 4·8-s + 6·9-s + 9·12-s − 2·13-s − 10·14-s + 5·16-s + 6·17-s − 12·18-s − 4·19-s + 15·21-s − 12·24-s + 7·25-s + 4·26-s + 9·27-s + 15·28-s + 16·31-s − 6·32-s − 12·34-s + 18·36-s + 8·38-s − 6·39-s − 30·42-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.73·3-s + 3/2·4-s − 2.44·6-s + 1.88·7-s − 1.41·8-s + 2·9-s + 2.59·12-s − 0.554·13-s − 2.67·14-s + 5/4·16-s + 1.45·17-s − 2.82·18-s − 0.917·19-s + 3.27·21-s − 2.44·24-s + 7/5·25-s + 0.784·26-s + 1.73·27-s + 2.83·28-s + 2.87·31-s − 1.06·32-s − 2.05·34-s + 3·36-s + 1.29·38-s − 0.960·39-s − 4.62·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.462734889\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.462734889\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−10.63307906535151344877941148036, −10.46376526543150581080534416424, −9.993640055714420937476191524988, −9.763433257296843544727887501009, −9.031067210053022948242216640783, −8.722226224748466288973523342383, −8.294591813681307967133182517608, −8.094264662643499913271655057725, −7.79980524658421591786138110960, −7.34759795774640514675833094177, −6.63372367973846661892352533072, −6.45468674211652865082649486593, −5.22218210313872774471035786715, −4.99184287982164150228404543748, −4.30176490076805125108065211901, −3.58970735283900584534933268208, −2.68028987451300128794575691986, −2.61103883449288255431384539732, −1.52481229676053942362246497882, −1.28885194920184347838577145624,
1.28885194920184347838577145624, 1.52481229676053942362246497882, 2.61103883449288255431384539732, 2.68028987451300128794575691986, 3.58970735283900584534933268208, 4.30176490076805125108065211901, 4.99184287982164150228404543748, 5.22218210313872774471035786715, 6.45468674211652865082649486593, 6.63372367973846661892352533072, 7.34759795774640514675833094177, 7.79980524658421591786138110960, 8.094264662643499913271655057725, 8.294591813681307967133182517608, 8.722226224748466288973523342383, 9.031067210053022948242216640783, 9.763433257296843544727887501009, 9.993640055714420937476191524988, 10.46376526543150581080534416424, 10.63307906535151344877941148036