| L(s) = 1 | + 2-s + 2.44·3-s + 4-s − 1.97·5-s + 2.44·6-s − 1.80·7-s + 8-s + 2.97·9-s − 1.97·10-s + 0.774·11-s + 2.44·12-s − 4.35·13-s − 1.80·14-s − 4.83·15-s + 16-s − 1.48·17-s + 2.97·18-s + 19-s − 1.97·20-s − 4.41·21-s + 0.774·22-s − 1.62·23-s + 2.44·24-s − 1.08·25-s − 4.35·26-s − 0.0536·27-s − 1.80·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.41·3-s + 0.5·4-s − 0.885·5-s + 0.998·6-s − 0.682·7-s + 0.353·8-s + 0.992·9-s − 0.625·10-s + 0.233·11-s + 0.705·12-s − 1.20·13-s − 0.482·14-s − 1.24·15-s + 0.250·16-s − 0.359·17-s + 0.701·18-s + 0.229·19-s − 0.442·20-s − 0.963·21-s + 0.165·22-s − 0.339·23-s + 0.499·24-s − 0.216·25-s − 0.854·26-s − 0.0103·27-s − 0.341·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 + 1.97T + 5T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 - 0.774T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 + 0.380T + 41T^{2} \) |
| 43 | \( 1 + 5.09T + 43T^{2} \) |
| 47 | \( 1 + 5.06T + 47T^{2} \) |
| 53 | \( 1 + 2.03T + 53T^{2} \) |
| 59 | \( 1 + 5.98T + 59T^{2} \) |
| 61 | \( 1 + 6.74T + 61T^{2} \) |
| 67 | \( 1 - 6.96T + 67T^{2} \) |
| 71 | \( 1 + 6.40T + 71T^{2} \) |
| 73 | \( 1 + 5.49T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 9.67T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.56746708298463994853690309741, −6.83670289943717669277499137884, −6.31629816299279394291577267917, −5.12428666647134122597401963192, −4.44524853166979538979428747413, −3.81591266731287334517016043617, −3.03374618406614967387905254658, −2.69879521732599959270286193877, −1.63425692909703502647080694375, 0,
1.63425692909703502647080694375, 2.69879521732599959270286193877, 3.03374618406614967387905254658, 3.81591266731287334517016043617, 4.44524853166979538979428747413, 5.12428666647134122597401963192, 6.31629816299279394291577267917, 6.83670289943717669277499137884, 7.56746708298463994853690309741
