| L(s) = 1 | + 2-s − 2.74·3-s + 4-s + 0.116·5-s − 2.74·6-s − 0.997·7-s + 8-s + 4.51·9-s + 0.116·10-s + 3.52·11-s − 2.74·12-s + 1.83·13-s − 0.997·14-s − 0.320·15-s + 16-s − 2.74·17-s + 4.51·18-s + 19-s + 0.116·20-s + 2.73·21-s + 3.52·22-s + 3.51·23-s − 2.74·24-s − 4.98·25-s + 1.83·26-s − 4.14·27-s − 0.997·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.58·3-s + 0.5·4-s + 0.0523·5-s − 1.11·6-s − 0.376·7-s + 0.353·8-s + 1.50·9-s + 0.0369·10-s + 1.06·11-s − 0.791·12-s + 0.509·13-s − 0.266·14-s − 0.0827·15-s + 0.250·16-s − 0.666·17-s + 1.06·18-s + 0.229·19-s + 0.0261·20-s + 0.596·21-s + 0.751·22-s + 0.732·23-s − 0.559·24-s − 0.997·25-s + 0.360·26-s − 0.797·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 - 0.116T + 5T^{2} \) |
| 7 | \( 1 + 0.997T + 7T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 17 | \( 1 + 2.74T + 17T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 + 2.84T + 29T^{2} \) |
| 31 | \( 1 + 7.76T + 31T^{2} \) |
| 37 | \( 1 - 1.28T + 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 1.96T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 2.10T + 79T^{2} \) |
| 83 | \( 1 - 0.895T + 83T^{2} \) |
| 89 | \( 1 - 4.04T + 89T^{2} \) |
| 97 | \( 1 - 2.54T + 97T^{2} \) |
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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
−6.94832058691111062537792795853, −6.61360227817784836126547784912, −6.17804955613567866009096635333, −5.20154674099959508198508555567, −5.05642831073251120249652456434, −3.83336435033402837276038832027, −3.58322125350961565825471623410, −2.08094773754668386852956904894, −1.22160400724322805009456388189, 0,
1.22160400724322805009456388189, 2.08094773754668386852956904894, 3.58322125350961565825471623410, 3.83336435033402837276038832027, 5.05642831073251120249652456434, 5.20154674099959508198508555567, 6.17804955613567866009096635333, 6.61360227817784836126547784912, 6.94832058691111062537792795853
