| L(s) = 1 | − 1.30·5-s + 1.64·7-s + 0.478·11-s + 2.76·13-s + 1.45·17-s + 19-s − 5.70·23-s − 3.29·25-s − 0.975·29-s + 5.09·31-s − 2.14·35-s − 9.70·37-s + 2.42·41-s − 5.99·43-s − 9.69·47-s − 4.30·49-s − 13.9·53-s − 0.625·55-s − 9.68·59-s + 14.8·61-s − 3.60·65-s + 0.708·67-s + 10.5·71-s + 4.57·73-s + 0.785·77-s + 13.0·79-s − 3.10·83-s + ⋯ |
| L(s) = 1 | − 0.584·5-s + 0.620·7-s + 0.144·11-s + 0.765·13-s + 0.352·17-s + 0.229·19-s − 1.19·23-s − 0.658·25-s − 0.181·29-s + 0.914·31-s − 0.362·35-s − 1.59·37-s + 0.379·41-s − 0.914·43-s − 1.41·47-s − 0.615·49-s − 1.92·53-s − 0.0843·55-s − 1.26·59-s + 1.90·61-s − 0.447·65-s + 0.0865·67-s + 1.24·71-s + 0.535·73-s + 0.0895·77-s + 1.46·79-s − 0.340·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 - 0.478T + 11T^{2} \) |
| 13 | \( 1 - 2.76T + 13T^{2} \) |
| 17 | \( 1 - 1.45T + 17T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 0.975T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 + 5.99T + 43T^{2} \) |
| 47 | \( 1 + 9.69T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 + 9.68T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 0.708T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 3.10T + 83T^{2} \) |
| 89 | \( 1 - 4.84T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.72268557998080069648181259832, −6.69390894750952895529573707126, −6.20537855646589918068005933981, −5.27342150964824562636833195297, −4.68690312393325528506854621237, −3.76625632037620699714960863376, −3.34894333007464121977701112242, −2.08026031244702027965478926524, −1.31876459242242034221258203212, 0,
1.31876459242242034221258203212, 2.08026031244702027965478926524, 3.34894333007464121977701112242, 3.76625632037620699714960863376, 4.68690312393325528506854621237, 5.27342150964824562636833195297, 6.20537855646589918068005933981, 6.69390894750952895529573707126, 7.72268557998080069648181259832
