| L(s) = 1 | + 1.41·5-s + 1.41·7-s + 2·11-s − 2·17-s − 4·19-s − 2.82·23-s − 2.99·25-s − 9.89·29-s − 7.07·31-s + 2.00·35-s − 8.48·37-s − 6·41-s − 8·43-s − 2.82·47-s − 5·49-s + 1.41·53-s + 2.82·55-s + 12·59-s + 14.1·61-s − 8·67-s + 14.1·71-s − 8·73-s + 2.82·77-s − 4.24·79-s + 6·83-s − 2.82·85-s − 2·89-s + ⋯ |
| L(s) = 1 | + 0.632·5-s + 0.534·7-s + 0.603·11-s − 0.485·17-s − 0.917·19-s − 0.589·23-s − 0.599·25-s − 1.83·29-s − 1.27·31-s + 0.338·35-s − 1.39·37-s − 0.937·41-s − 1.21·43-s − 0.412·47-s − 0.714·49-s + 0.194·53-s + 0.381·55-s + 1.56·59-s + 1.81·61-s − 0.977·67-s + 1.67·71-s − 0.936·73-s + 0.322·77-s − 0.477·79-s + 0.658·83-s − 0.306·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.083400092509309447820919807206, −7.09380594818878033884864566768, −6.58216225485557033095722837256, −5.65796765002052045802713295943, −5.15977761347797516334687653687, −4.09271876452117206942159769554, −3.50878290635458755194079886524, −2.01235288403696179765377395125, −1.77329287335247602298611789780, 0,
1.77329287335247602298611789780, 2.01235288403696179765377395125, 3.50878290635458755194079886524, 4.09271876452117206942159769554, 5.15977761347797516334687653687, 5.65796765002052045802713295943, 6.58216225485557033095722837256, 7.09380594818878033884864566768, 8.083400092509309447820919807206
