| L(s) = 1 | − 2.74·2-s + 2.10·3-s + 5.54·4-s + 3.39·5-s − 5.79·6-s − 0.0705·7-s − 9.74·8-s + 1.45·9-s − 9.31·10-s − 4.86·11-s + 11.7·12-s + 2.06·13-s + 0.193·14-s + 7.15·15-s + 15.6·16-s − 2.52·17-s − 3.98·18-s − 5.74·19-s + 18.8·20-s − 0.148·21-s + 13.3·22-s + 6.77·23-s − 20.5·24-s + 6.49·25-s − 5.68·26-s − 3.26·27-s − 0.391·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 1.21·3-s + 2.77·4-s + 1.51·5-s − 2.36·6-s − 0.0266·7-s − 3.44·8-s + 0.483·9-s − 2.94·10-s − 1.46·11-s + 3.37·12-s + 0.573·13-s + 0.0518·14-s + 1.84·15-s + 3.92·16-s − 0.612·17-s − 0.939·18-s − 1.31·19-s + 4.20·20-s − 0.0324·21-s + 2.84·22-s + 1.41·23-s − 4.19·24-s + 1.29·25-s − 1.11·26-s − 0.629·27-s − 0.0739·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 - 3.39T + 5T^{2} \) |
| 7 | \( 1 + 0.0705T + 7T^{2} \) |
| 11 | \( 1 + 4.86T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 19 | \( 1 + 5.74T + 19T^{2} \) |
| 23 | \( 1 - 6.77T + 23T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 - 5.42T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 7.49T + 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 6.07T + 53T^{2} \) |
| 59 | \( 1 + 4.42T + 59T^{2} \) |
| 61 | \( 1 + 7.53T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 9.27T + 79T^{2} \) |
| 83 | \( 1 + 5.20T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.84962772009304103525926127246, −7.06064199218653745973677926216, −6.33054453299376766040801964051, −5.85676000356556139519660142124, −4.78350252288195893508422247806, −3.11692937083432987676904503634, −2.75904726414063340841950864336, −2.02513223967374940954284552019, −1.48968412647830317674959854595, 0,
1.48968412647830317674959854595, 2.02513223967374940954284552019, 2.75904726414063340841950864336, 3.11692937083432987676904503634, 4.78350252288195893508422247806, 5.85676000356556139519660142124, 6.33054453299376766040801964051, 7.06064199218653745973677926216, 7.84962772009304103525926127246
