| L(s) = 1 | + 2-s + 0.180·3-s + 4-s + 0.229·5-s + 0.180·6-s − 1.11·7-s + 8-s − 2.96·9-s + 0.229·10-s − 4.26·11-s + 0.180·12-s + 5.52·13-s − 1.11·14-s + 0.0413·15-s + 16-s + 3.05·17-s − 2.96·18-s + 19-s + 0.229·20-s − 0.200·21-s − 4.26·22-s − 0.559·23-s + 0.180·24-s − 4.94·25-s + 5.52·26-s − 1.07·27-s − 1.11·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.104·3-s + 0.5·4-s + 0.102·5-s + 0.0735·6-s − 0.420·7-s + 0.353·8-s − 0.989·9-s + 0.0726·10-s − 1.28·11-s + 0.0520·12-s + 1.53·13-s − 0.297·14-s + 0.0106·15-s + 0.250·16-s + 0.742·17-s − 0.699·18-s + 0.229·19-s + 0.0513·20-s − 0.0437·21-s − 0.910·22-s − 0.116·23-s + 0.0367·24-s − 0.989·25-s + 1.08·26-s − 0.206·27-s − 0.210·28-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 | \( 1 - 0.180T + 3T^{2} \) |
| 5 | \( 1 - 0.229T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 - 5.52T + 13T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 23 | \( 1 + 0.559T + 23T^{2} \) |
| 29 | \( 1 + 4.00T + 29T^{2} \) |
| 31 | \( 1 - 9.37T + 31T^{2} \) |
| 37 | \( 1 - 4.95T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 + 0.623T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 9.26T + 67T^{2} \) |
| 71 | \( 1 - 8.20T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 4.06T + 79T^{2} \) |
| 83 | \( 1 - 4.31T + 83T^{2} \) |
| 89 | \( 1 + 2.30T + 89T^{2} \) |
| 97 | \( 1 + 2.67T + 97T^{2} \) |
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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
−7.60235683473938237744484290531, −6.34173764900672628078816200413, −6.23130207886353243573450284315, −5.34530419812431020094314947335, −4.86024921638289837347529066542, −3.61109281255887254125696433510, −3.29266479686481312949635336471, −2.48069841055599003491785089902, −1.43508431479748758931349643959, 0,
1.43508431479748758931349643959, 2.48069841055599003491785089902, 3.29266479686481312949635336471, 3.61109281255887254125696433510, 4.86024921638289837347529066542, 5.34530419812431020094314947335, 6.23130207886353243573450284315, 6.34173764900672628078816200413, 7.60235683473938237744484290531
