| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s − 11-s + 14-s − 16-s − 2·17-s − 4·19-s + 22-s + 8·23-s − 5·25-s + 28-s + 4·29-s − 8·31-s − 5·32-s + 2·34-s − 8·37-s + 4·38-s + 2·41-s − 12·43-s + 44-s − 8·46-s + 2·47-s + 49-s + 5·50-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s − 0.301·11-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.213·22-s + 1.66·23-s − 25-s + 0.188·28-s + 0.742·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s − 1.31·37-s + 0.648·38-s + 0.312·41-s − 1.82·43-s + 0.150·44-s − 1.17·46-s + 0.291·47-s + 1/7·49-s + 0.707·50-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.83523594618205, −13.16080253317226, −12.96257258425553, −12.60164143676768, −11.81274748776456, −11.20903808290834, −10.88800815161229, −10.27787109771662, −9.945358148489172, −9.379562051340629, −8.922299027686120, −8.436935720648158, −8.173921585533833, −7.363276869307553, −6.899100244094035, −6.609124292410996, −5.597050002573077, −5.312806385242332, −4.662048501032058, −4.104223720360899, −3.532360686390833, −2.904478984177540, −2.043695158384828, −1.570932729611320, −0.6266084764358323, 0,
0.6266084764358323, 1.570932729611320, 2.043695158384828, 2.904478984177540, 3.532360686390833, 4.104223720360899, 4.662048501032058, 5.312806385242332, 5.597050002573077, 6.609124292410996, 6.899100244094035, 7.363276869307553, 8.173921585533833, 8.436935720648158, 8.922299027686120, 9.379562051340629, 9.945358148489172, 10.27787109771662, 10.88800815161229, 11.20903808290834, 11.81274748776456, 12.60164143676768, 12.96257258425553, 13.16080253317226, 13.83523594618205
