| L(s) = 1 | + 1.06·2-s − 3-s − 0.869·4-s + 2.24·5-s − 1.06·6-s − 7-s − 3.05·8-s + 9-s + 2.38·10-s − 4.08·11-s + 0.869·12-s + 6.83·13-s − 1.06·14-s − 2.24·15-s − 1.50·16-s − 1.80·17-s + 1.06·18-s − 6.43·19-s − 1.95·20-s + 21-s − 4.34·22-s − 5.39·23-s + 3.05·24-s + 0.0495·25-s + 7.26·26-s − 27-s + 0.869·28-s + ⋯ |
| L(s) = 1 | + 0.751·2-s − 0.577·3-s − 0.434·4-s + 1.00·5-s − 0.434·6-s − 0.377·7-s − 1.07·8-s + 0.333·9-s + 0.755·10-s − 1.23·11-s + 0.251·12-s + 1.89·13-s − 0.284·14-s − 0.580·15-s − 0.376·16-s − 0.437·17-s + 0.250·18-s − 1.47·19-s − 0.436·20-s + 0.218·21-s − 0.926·22-s − 1.12·23-s + 0.622·24-s + 0.00990·25-s + 1.42·26-s − 0.192·27-s + 0.164·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.688318977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.688318977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 11 | \( 1 + 4.08T + 11T^{2} \) |
| 13 | \( 1 - 6.83T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 + 6.43T + 19T^{2} \) |
| 23 | \( 1 + 5.39T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 - 4.71T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 4.74T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 8.39T + 71T^{2} \) |
| 73 | \( 1 - 4.30T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 6.23T + 97T^{2} \) |
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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
−7.978342238179362603762139068486, −6.69051279435972041602988216578, −6.08249723911491713012337959075, −5.86622580706931575826357192605, −5.18596104063977001186550616847, −4.21675341853275020658733824246, −3.83179948494620663429168210375, −2.69055753947991364692315686820, −1.93700445709017593440195109501, −0.57437524423329004025475756672,
0.57437524423329004025475756672, 1.93700445709017593440195109501, 2.69055753947991364692315686820, 3.83179948494620663429168210375, 4.21675341853275020658733824246, 5.18596104063977001186550616847, 5.86622580706931575826357192605, 6.08249723911491713012337959075, 6.69051279435972041602988216578, 7.978342238179362603762139068486
