| L(s) = 1 | − 2-s + 4-s + 3.23·5-s + 3.10·7-s − 8-s − 3.23·10-s + 0.565·11-s − 0.755·13-s − 3.10·14-s + 16-s + 0.292·17-s + 3.68·19-s + 3.23·20-s − 0.565·22-s + 5.90·23-s + 5.44·25-s + 0.755·26-s + 3.10·28-s + 4.02·29-s + 2.64·31-s − 32-s − 0.292·34-s + 10.0·35-s + 4.18·37-s − 3.68·38-s − 3.23·40-s − 6.81·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.44·5-s + 1.17·7-s − 0.353·8-s − 1.02·10-s + 0.170·11-s − 0.209·13-s − 0.830·14-s + 0.250·16-s + 0.0709·17-s + 0.844·19-s + 0.722·20-s − 0.120·22-s + 1.23·23-s + 1.08·25-s + 0.148·26-s + 0.587·28-s + 0.747·29-s + 0.474·31-s − 0.176·32-s − 0.0501·34-s + 1.69·35-s + 0.687·37-s − 0.597·38-s − 0.510·40-s − 1.06·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.703084322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.703084322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 - 0.565T + 11T^{2} \) |
| 13 | \( 1 + 0.755T + 13T^{2} \) |
| 17 | \( 1 - 0.292T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 - 4.02T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 4.18T + 37T^{2} \) |
| 41 | \( 1 + 6.81T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 - 4.40T + 47T^{2} \) |
| 53 | \( 1 + 7.88T + 53T^{2} \) |
| 59 | \( 1 - 1.95T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 0.800T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 2.54T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.81005356876209622421905133687, −7.28986128670264632565315694071, −6.43189247368225842554914329592, −5.85242809389671947269548482948, −5.08077186346519934775689010049, −4.57615949547952838544190293982, −3.17424154189309763591160943830, −2.44803673282831249881912267065, −1.59509451168848813044762977077, −1.00865231182213429786280207463,
1.00865231182213429786280207463, 1.59509451168848813044762977077, 2.44803673282831249881912267065, 3.17424154189309763591160943830, 4.57615949547952838544190293982, 5.08077186346519934775689010049, 5.85242809389671947269548482948, 6.43189247368225842554914329592, 7.28986128670264632565315694071, 7.81005356876209622421905133687
