| L(s) = 1 | − 2-s − 2·5-s + 8-s + 3·9-s + 2·10-s + 11-s + 4·13-s − 16-s − 2·17-s − 3·18-s − 22-s + 8·23-s + 5·25-s − 4·26-s − 4·29-s + 8·31-s + 2·34-s + 2·37-s − 2·40-s + 20·41-s + 8·43-s − 6·45-s − 8·46-s − 8·47-s − 5·50-s − 6·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s + 0.353·8-s + 9-s + 0.632·10-s + 0.301·11-s + 1.10·13-s − 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.213·22-s + 1.66·23-s + 25-s − 0.784·26-s − 0.742·29-s + 1.43·31-s + 0.342·34-s + 0.328·37-s − 0.316·40-s + 3.12·41-s + 1.21·43-s − 0.894·45-s − 1.17·46-s − 1.16·47-s − 0.707·50-s − 0.824·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.631261560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631261560\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.866680128154662263462566959207, −9.624130535084563857105116220844, −9.114758760581381904312874453900, −9.087765083059639597812593902775, −8.357984381276078225586116532786, −8.044698259121380828699230515823, −7.72415773165913018780646671439, −7.28250147827492115171558680778, −6.80634681628425936349539485406, −6.41479687663417399200876361941, −6.08806478095506394916109331882, −5.23118869012221395816040035325, −4.79551662724391612437295657993, −4.38641952433449779125522636403, −3.87708234342747849797420863336, −3.51839276428916538272347388440, −2.75724397740622672694641407672, −2.11151157087051574290647776856, −0.982740643493707664849252053046, −0.927084690643328753274381074031,
0.927084690643328753274381074031, 0.982740643493707664849252053046, 2.11151157087051574290647776856, 2.75724397740622672694641407672, 3.51839276428916538272347388440, 3.87708234342747849797420863336, 4.38641952433449779125522636403, 4.79551662724391612437295657993, 5.23118869012221395816040035325, 6.08806478095506394916109331882, 6.41479687663417399200876361941, 6.80634681628425936349539485406, 7.28250147827492115171558680778, 7.72415773165913018780646671439, 8.044698259121380828699230515823, 8.357984381276078225586116532786, 9.087765083059639597812593902775, 9.114758760581381904312874453900, 9.624130535084563857105116220844, 9.866680128154662263462566959207
