| L(s) = 1 | + 2-s − 1.08·3-s + 4-s + 0.379·5-s − 1.08·6-s + 0.614·7-s + 8-s − 1.83·9-s + 0.379·10-s + 5.27·11-s − 1.08·12-s − 0.100·13-s + 0.614·14-s − 0.409·15-s + 16-s − 4.27·17-s − 1.83·18-s − 3.59·19-s + 0.379·20-s − 0.664·21-s + 5.27·22-s − 23-s − 1.08·24-s − 4.85·25-s − 0.100·26-s + 5.22·27-s + 0.614·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.624·3-s + 0.5·4-s + 0.169·5-s − 0.441·6-s + 0.232·7-s + 0.353·8-s − 0.610·9-s + 0.119·10-s + 1.59·11-s − 0.312·12-s − 0.0278·13-s + 0.164·14-s − 0.105·15-s + 0.250·16-s − 1.03·17-s − 0.431·18-s − 0.823·19-s + 0.0847·20-s − 0.145·21-s + 1.12·22-s − 0.208·23-s − 0.220·24-s − 0.971·25-s − 0.0196·26-s + 1.00·27-s + 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 - 0.379T + 5T^{2} \) |
| 7 | \( 1 - 0.614T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 + 0.100T + 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 8.15T + 41T^{2} \) |
| 43 | \( 1 + 7.73T + 43T^{2} \) |
| 47 | \( 1 + 8.71T + 47T^{2} \) |
| 53 | \( 1 - 1.61T + 53T^{2} \) |
| 59 | \( 1 + 7.31T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 3.66T + 71T^{2} \) |
| 73 | \( 1 - 2.16T + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 0.0472T + 89T^{2} \) |
| 97 | \( 1 - 0.462T + 97T^{2} \) |
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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
−7.58262078313134141844649958338, −6.64602200853062095384638306293, −6.24270433480884679355650850442, −5.70592937493101996507334079058, −4.76823650841447594509992539103, −4.16677934466013530960370651139, −3.43599607979843622876753534004, −2.28058011599477486477273898237, −1.51051422689392071181072203259, 0,
1.51051422689392071181072203259, 2.28058011599477486477273898237, 3.43599607979843622876753534004, 4.16677934466013530960370651139, 4.76823650841447594509992539103, 5.70592937493101996507334079058, 6.24270433480884679355650850442, 6.64602200853062095384638306293, 7.58262078313134141844649958338
