| L(s) = 1 | + 1.39·2-s − 2.59·3-s − 0.0526·4-s − 3.62·6-s + 1.49·7-s − 2.86·8-s + 3.75·9-s − 1.55·11-s + 0.136·12-s − 0.546·13-s + 2.08·14-s − 3.89·16-s + 6.16·17-s + 5.24·18-s − 4.52·19-s − 3.88·21-s − 2.17·22-s + 0.734·23-s + 7.44·24-s − 0.763·26-s − 1.97·27-s − 0.0788·28-s + 3.56·29-s − 4.59·31-s + 0.298·32-s + 4.04·33-s + 8.60·34-s + ⋯ |
| L(s) = 1 | + 0.986·2-s − 1.50·3-s − 0.0263·4-s − 1.48·6-s + 0.565·7-s − 1.01·8-s + 1.25·9-s − 0.469·11-s + 0.0395·12-s − 0.151·13-s + 0.557·14-s − 0.972·16-s + 1.49·17-s + 1.23·18-s − 1.03·19-s − 0.848·21-s − 0.463·22-s + 0.153·23-s + 1.52·24-s − 0.149·26-s − 0.379·27-s − 0.0148·28-s + 0.662·29-s − 0.826·31-s + 0.0526·32-s + 0.704·33-s + 1.47·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 + 2.59T + 3T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 + 1.55T + 11T^{2} \) |
| 13 | \( 1 + 0.546T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 + 4.52T + 19T^{2} \) |
| 23 | \( 1 - 0.734T + 23T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 + 4.59T + 31T^{2} \) |
| 37 | \( 1 + 2.22T + 37T^{2} \) |
| 41 | \( 1 - 3.66T + 41T^{2} \) |
| 43 | \( 1 - 1.59T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 - 9.96T + 53T^{2} \) |
| 59 | \( 1 - 7.91T + 59T^{2} \) |
| 61 | \( 1 + 5.49T + 61T^{2} \) |
| 67 | \( 1 - 9.98T + 67T^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 9.19T + 83T^{2} \) |
| 89 | \( 1 - 7.28T + 89T^{2} \) |
| 97 | \( 1 - 6.07T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50271987485010813761421064513, −6.72626315107821626029863846914, −5.92059813048479387346723700077, −5.54128579768539991937090627468, −4.93705444402655721335897266405, −4.36009146986391686788993575804, −3.51996366446914910871352350929, −2.46412121453526570601997138549, −1.13090084312646799316111289781, 0,
1.13090084312646799316111289781, 2.46412121453526570601997138549, 3.51996366446914910871352350929, 4.36009146986391686788993575804, 4.93705444402655721335897266405, 5.54128579768539991937090627468, 5.92059813048479387346723700077, 6.72626315107821626029863846914, 7.50271987485010813761421064513
