| L(s) = 1 | + 3-s + 3.17·5-s − 0.651·7-s + 9-s − 2.35·11-s + 5.77·13-s + 3.17·15-s − 0.736·17-s + 3.04·19-s − 0.651·21-s + 1.45·23-s + 5.10·25-s + 27-s + 3.59·29-s − 5.65·31-s − 2.35·33-s − 2.07·35-s + 7.15·37-s + 5.77·39-s + 8.85·41-s + 4.96·43-s + 3.17·45-s − 3.18·47-s − 6.57·49-s − 0.736·51-s + 1.50·53-s − 7.48·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.42·5-s − 0.246·7-s + 0.333·9-s − 0.710·11-s + 1.60·13-s + 0.820·15-s − 0.178·17-s + 0.699·19-s − 0.142·21-s + 0.303·23-s + 1.02·25-s + 0.192·27-s + 0.667·29-s − 1.01·31-s − 0.410·33-s − 0.350·35-s + 1.17·37-s + 0.925·39-s + 1.38·41-s + 0.757·43-s + 0.473·45-s − 0.463·47-s − 0.939·49-s − 0.103·51-s + 0.206·53-s − 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.802787470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.802787470\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.17T + 5T^{2} \) |
| 7 | \( 1 + 0.651T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 5.77T + 13T^{2} \) |
| 17 | \( 1 + 0.736T + 17T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 23 | \( 1 - 1.45T + 23T^{2} \) |
| 29 | \( 1 - 3.59T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 7.15T + 37T^{2} \) |
| 41 | \( 1 - 8.85T + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 + 3.18T + 47T^{2} \) |
| 53 | \( 1 - 1.50T + 53T^{2} \) |
| 59 | \( 1 + 3.09T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 6.06T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 + 1.00T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 8.30T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 9.62T + 97T^{2} \) |
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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
−7.893845864500851963611175170817, −7.16747306297230020876064603585, −6.20200081268935148176485756096, −5.96073448898044910682063401756, −5.14159811007949523100525019551, −4.26074509102039710091784804739, −3.29690874335768691552042573860, −2.68684941155276941042052333811, −1.82076957964637101252894348918, −0.996345833357379752189259597592,
0.996345833357379752189259597592, 1.82076957964637101252894348918, 2.68684941155276941042052333811, 3.29690874335768691552042573860, 4.26074509102039710091784804739, 5.14159811007949523100525019551, 5.96073448898044910682063401756, 6.20200081268935148176485756096, 7.16747306297230020876064603585, 7.893845864500851963611175170817
