| L(s) = 1 | + 3-s − 1.48·5-s − 1.03·7-s + 9-s + 5.64·11-s − 4.47·13-s − 1.48·15-s − 6.13·17-s + 5.87·19-s − 1.03·21-s − 1.22·23-s − 2.80·25-s + 27-s − 6.91·29-s + 3.70·31-s + 5.64·33-s + 1.54·35-s − 6.62·37-s − 4.47·39-s + 1.85·41-s − 2.29·43-s − 1.48·45-s + 3.59·47-s − 5.92·49-s − 6.13·51-s − 0.299·53-s − 8.36·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.662·5-s − 0.392·7-s + 0.333·9-s + 1.70·11-s − 1.24·13-s − 0.382·15-s − 1.48·17-s + 1.34·19-s − 0.226·21-s − 0.255·23-s − 0.560·25-s + 0.192·27-s − 1.28·29-s + 0.664·31-s + 0.981·33-s + 0.260·35-s − 1.08·37-s − 0.717·39-s + 0.290·41-s − 0.350·43-s − 0.220·45-s + 0.523·47-s − 0.845·49-s − 0.859·51-s − 0.0411·53-s − 1.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 6.13T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 + 1.22T + 23T^{2} \) |
| 29 | \( 1 + 6.91T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 + 6.62T + 37T^{2} \) |
| 41 | \( 1 - 1.85T + 41T^{2} \) |
| 43 | \( 1 + 2.29T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 + 0.299T + 53T^{2} \) |
| 59 | \( 1 - 0.102T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 - 2.00T + 71T^{2} \) |
| 73 | \( 1 - 9.34T + 73T^{2} \) |
| 79 | \( 1 + 3.92T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 3.40T + 97T^{2} \) |
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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
−8.119884888349185442392469794756, −7.15843039228053126184270824646, −6.97057949578169473694952149784, −5.95730780915483295830675783480, −4.88572829276751503110756923526, −4.07944146284679489219247240254, −3.54870421830755294067481489400, −2.53260671561345394896616841098, −1.51092403242314701006793699660, 0,
1.51092403242314701006793699660, 2.53260671561345394896616841098, 3.54870421830755294067481489400, 4.07944146284679489219247240254, 4.88572829276751503110756923526, 5.95730780915483295830675783480, 6.97057949578169473694952149784, 7.15843039228053126184270824646, 8.119884888349185442392469794756
