L(s) = 1 | + 1.17·2-s + 3-s − 0.629·4-s + 0.418·5-s + 1.17·6-s + 1.98·7-s − 3.07·8-s + 9-s + 0.490·10-s − 5.41·11-s − 0.629·12-s − 1.43·13-s + 2.32·14-s + 0.418·15-s − 2.34·16-s − 6.03·17-s + 1.17·18-s − 6.20·19-s − 0.263·20-s + 1.98·21-s − 6.33·22-s − 23-s − 3.07·24-s − 4.82·25-s − 1.68·26-s + 27-s − 1.24·28-s + ⋯ |
L(s) = 1 | + 0.827·2-s + 0.577·3-s − 0.314·4-s + 0.187·5-s + 0.478·6-s + 0.748·7-s − 1.08·8-s + 0.333·9-s + 0.155·10-s − 1.63·11-s − 0.181·12-s − 0.399·13-s + 0.620·14-s + 0.108·15-s − 0.586·16-s − 1.46·17-s + 0.275·18-s − 1.42·19-s − 0.0589·20-s + 0.432·21-s − 1.35·22-s − 0.208·23-s − 0.628·24-s − 0.964·25-s − 0.330·26-s + 0.192·27-s − 0.235·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 5 | \( 1 - 0.418T + 5T^{2} \) |
| 7 | \( 1 - 1.98T + 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 + 1.43T + 13T^{2} \) |
| 17 | \( 1 + 6.03T + 17T^{2} \) |
| 19 | \( 1 + 6.20T + 19T^{2} \) |
| 31 | \( 1 + 6.35T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 8.27T + 41T^{2} \) |
| 43 | \( 1 - 6.28T + 43T^{2} \) |
| 47 | \( 1 - 6.99T + 47T^{2} \) |
| 53 | \( 1 - 5.10T + 53T^{2} \) |
| 59 | \( 1 - 9.51T + 59T^{2} \) |
| 61 | \( 1 + 4.79T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 7.30T + 71T^{2} \) |
| 73 | \( 1 + 4.61T + 73T^{2} \) |
| 79 | \( 1 + 4.07T + 79T^{2} \) |
| 83 | \( 1 - 3.71T + 83T^{2} \) |
| 89 | \( 1 + 0.547T + 89T^{2} \) |
| 97 | \( 1 + 8.58T + 97T^{2} \) |
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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
−8.752116622708251397889983853715, −8.026553781665255769515691803386, −7.31542004883182494986005000755, −6.11610945180267226025088199318, −5.43133218433426038531897130407, −4.48977406172041110571684228508, −4.10899390878836019766944822112, −2.67389853080316596280155605773, −2.16668558272558830836989764920, 0,
2.16668558272558830836989764920, 2.67389853080316596280155605773, 4.10899390878836019766944822112, 4.48977406172041110571684228508, 5.43133218433426038531897130407, 6.11610945180267226025088199318, 7.31542004883182494986005000755, 8.026553781665255769515691803386, 8.752116622708251397889983853715