| L(s) = 1 | + 2-s + 0.347·3-s + 4-s + 0.347·6-s − 7-s + 8-s − 2.87·9-s + 0.532·11-s + 0.347·12-s − 1.22·13-s − 14-s + 16-s − 17-s − 2.87·18-s + 5.47·19-s − 0.347·21-s + 0.532·22-s + 4.10·23-s + 0.347·24-s − 1.22·26-s − 2.04·27-s − 28-s − 9.70·29-s − 5.78·31-s + 32-s + 0.184·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.200·3-s + 0.5·4-s + 0.141·6-s − 0.377·7-s + 0.353·8-s − 0.959·9-s + 0.160·11-s + 0.100·12-s − 0.340·13-s − 0.267·14-s + 0.250·16-s − 0.242·17-s − 0.678·18-s + 1.25·19-s − 0.0757·21-s + 0.113·22-s + 0.856·23-s + 0.0708·24-s − 0.240·26-s − 0.392·27-s − 0.188·28-s − 1.80·29-s − 1.03·31-s + 0.176·32-s + 0.0321·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.347T + 3T^{2} \) |
| 11 | \( 1 - 0.532T + 11T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 - 4.10T + 23T^{2} \) |
| 29 | \( 1 + 9.70T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 - 1.90T + 43T^{2} \) |
| 47 | \( 1 + 1.34T + 47T^{2} \) |
| 53 | \( 1 - 0.652T + 53T^{2} \) |
| 59 | \( 1 + 0.509T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 - 7.92T + 67T^{2} \) |
| 71 | \( 1 + 0.361T + 71T^{2} \) |
| 73 | \( 1 + 7.61T + 73T^{2} \) |
| 79 | \( 1 - 4.29T + 79T^{2} \) |
| 83 | \( 1 - 5.32T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 + 9.06T + 97T^{2} \) |
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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
−7.47889440014570384352141110427, −7.09792825591559306376584133891, −6.19098252513052596281360266702, −5.41098218999954821767448800019, −5.08271107262260990683912738561, −3.82866631443380196921255398255, −3.34122411823723706438154912050, −2.56150591513611287448079457468, −1.55813045773453920095586631881, 0,
1.55813045773453920095586631881, 2.56150591513611287448079457468, 3.34122411823723706438154912050, 3.82866631443380196921255398255, 5.08271107262260990683912738561, 5.41098218999954821767448800019, 6.19098252513052596281360266702, 7.09792825591559306376584133891, 7.47889440014570384352141110427
