| L(s) = 1 | − 2-s + 4-s − 4.41·5-s − 8-s + 4.41·10-s + 11-s + 16-s − 2.37·17-s − 3.68·19-s − 4.41·20-s − 22-s − 4.73·23-s + 14.5·25-s + 6.52·29-s − 3.37·31-s − 32-s + 2.37·34-s + 7.68·37-s + 3.68·38-s + 4.41·40-s − 12.1·41-s + 3.06·43-s + 44-s + 4.73·46-s + 4.10·47-s − 14.5·50-s − 8·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.97·5-s − 0.353·8-s + 1.39·10-s + 0.301·11-s + 0.250·16-s − 0.576·17-s − 0.846·19-s − 0.988·20-s − 0.213·22-s − 0.986·23-s + 2.90·25-s + 1.21·29-s − 0.606·31-s − 0.176·32-s + 0.407·34-s + 1.26·37-s + 0.598·38-s + 0.698·40-s − 1.89·41-s + 0.467·43-s + 0.150·44-s + 0.697·46-s + 0.599·47-s − 2.05·50-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 4.41T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 6.52T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 3.06T + 43T^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 3.79T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 0.934T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 0.418T + 79T^{2} \) |
| 83 | \( 1 - 2.14T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−7.53007439044715456806287020262, −6.75473704140712739250575622704, −6.38002215852065879665967483075, −5.16328095151499299345461209047, −4.32701290164631969254751161653, −3.88192433243789002706517155410, −3.06322225371137151667421357932, −2.12249707919965707724894924317, −0.862623496353935634237203731785, 0,
0.862623496353935634237203731785, 2.12249707919965707724894924317, 3.06322225371137151667421357932, 3.88192433243789002706517155410, 4.32701290164631969254751161653, 5.16328095151499299345461209047, 6.38002215852065879665967483075, 6.75473704140712739250575622704, 7.53007439044715456806287020262
