| L(s) = 1 | + 7-s + 2.64·11-s − 5.29·17-s + 7.93·23-s − 7.93·29-s − 10·31-s − 11·37-s − 10.5·41-s − 43-s + 10.5·47-s + 49-s − 10.5·53-s + 10.5·59-s − 8·61-s + 3·67-s − 2.64·71-s + 10·73-s + 2.64·77-s − 11·79-s − 5.29·83-s + 10.5·89-s − 2·97-s + 15.8·101-s + 8·103-s − 15.8·107-s + 3·109-s + 2.64·113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.797·11-s − 1.28·17-s + 1.65·23-s − 1.47·29-s − 1.79·31-s − 1.80·37-s − 1.65·41-s − 0.152·43-s + 1.54·47-s + 0.142·49-s − 1.45·53-s + 1.37·59-s − 1.02·61-s + 0.366·67-s − 0.313·71-s + 1.17·73-s + 0.301·77-s − 1.23·79-s − 0.580·83-s + 1.12·89-s − 0.203·97-s + 1.57·101-s + 0.788·103-s − 1.53·107-s + 0.287·109-s + 0.248·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 7.93T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 5.29T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.47296742311939075860670516664, −7.06069018890767508642474962665, −6.39728194182531935636266108157, −5.40061961017984438594880110464, −4.91748212917055359219610410262, −3.93378091201686525303816031148, −3.35886635460672454845798409320, −2.14667339897850451554152562933, −1.45046505865770407585813835701, 0,
1.45046505865770407585813835701, 2.14667339897850451554152562933, 3.35886635460672454845798409320, 3.93378091201686525303816031148, 4.91748212917055359219610410262, 5.40061961017984438594880110464, 6.39728194182531935636266108157, 7.06069018890767508642474962665, 7.47296742311939075860670516664
