| L(s) = 1 | + 3-s + 4.42·5-s − 0.622·7-s + 9-s + 5.80·11-s − 2·13-s + 4.42·15-s + 1.37·17-s + 6.42·19-s − 0.622·21-s + 23-s + 14.6·25-s + 27-s + 0.755·29-s + 1.24·31-s + 5.80·33-s − 2.75·35-s − 9.05·37-s − 2·39-s − 9.61·41-s − 5.18·43-s + 4.42·45-s + 5.24·47-s − 6.61·49-s + 1.37·51-s − 12.0·53-s + 25.7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.98·5-s − 0.235·7-s + 0.333·9-s + 1.75·11-s − 0.554·13-s + 1.14·15-s + 0.334·17-s + 1.47·19-s − 0.135·21-s + 0.208·23-s + 2.92·25-s + 0.192·27-s + 0.140·29-s + 0.223·31-s + 1.01·33-s − 0.465·35-s − 1.48·37-s − 0.320·39-s − 1.50·41-s − 0.790·43-s + 0.660·45-s + 0.764·47-s − 0.944·49-s + 0.192·51-s − 1.65·53-s + 3.46·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.108003711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.108003711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 4.42T + 5T^{2} \) |
| 7 | \( 1 + 0.622T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 - 6.42T + 19T^{2} \) |
| 29 | \( 1 - 0.755T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 + 5.18T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 - 5.18T + 67T^{2} \) |
| 71 | \( 1 + 6.10T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8.62T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 6.23T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−8.637303690490339776784803431702, −7.55527280894895366884682747507, −6.60183655505472391739538942184, −6.44367948805154453125882944983, −5.34909417191499042338381864691, −4.84520439998880821060633731084, −3.51671604833098938828319678058, −2.92700411661605780429272731554, −1.76900906091060646633401155993, −1.31660757416818293654552097499,
1.31660757416818293654552097499, 1.76900906091060646633401155993, 2.92700411661605780429272731554, 3.51671604833098938828319678058, 4.84520439998880821060633731084, 5.34909417191499042338381864691, 6.44367948805154453125882944983, 6.60183655505472391739538942184, 7.55527280894895366884682747507, 8.637303690490339776784803431702
