| L(s) = 1 | + 1.70·5-s − 0.630·7-s − 1.09·11-s + 2.34·13-s + 3.26·17-s + 19-s − 1.70·23-s − 2.07·25-s − 5.58·29-s − 8.12·31-s − 1.07·35-s + 3.12·37-s − 3.29·41-s − 7.36·43-s − 8.95·47-s − 6.60·49-s + 7.14·53-s − 1.86·55-s − 14.7·59-s − 11.2·61-s + 3.99·65-s + 7.04·67-s + 3.84·71-s − 1.29·73-s + 0.688·77-s + 7.68·79-s + 5.69·83-s + ⋯ |
| L(s) = 1 | + 0.764·5-s − 0.238·7-s − 0.329·11-s + 0.649·13-s + 0.791·17-s + 0.229·19-s − 0.356·23-s − 0.415·25-s − 1.03·29-s − 1.45·31-s − 0.182·35-s + 0.514·37-s − 0.515·41-s − 1.12·43-s − 1.30·47-s − 0.943·49-s + 0.980·53-s − 0.251·55-s − 1.92·59-s − 1.43·61-s + 0.496·65-s + 0.861·67-s + 0.456·71-s − 0.151·73-s + 0.0784·77-s + 0.864·79-s + 0.625·83-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 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 + 0.630T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 + 7.36T + 43T^{2} \) |
| 47 | \( 1 + 8.95T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 7.04T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 - 7.68T + 79T^{2} \) |
| 83 | \( 1 - 5.69T + 83T^{2} \) |
| 89 | \( 1 + 6.66T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−7.59381969511318517971316197462, −6.65387412102182874386990895870, −6.03460120275776525739652876075, −5.49128305432659667486148087163, −4.80679544151451295447773731458, −3.68029917526854921603989509069, −3.22787296952861829271559697311, −2.06929839612240596659386832195, −1.44299009555910472471794043313, 0,
1.44299009555910472471794043313, 2.06929839612240596659386832195, 3.22787296952861829271559697311, 3.68029917526854921603989509069, 4.80679544151451295447773731458, 5.49128305432659667486148087163, 6.03460120275776525739652876075, 6.65387412102182874386990895870, 7.59381969511318517971316197462
