| L(s) = 1 | − 2-s − 2.15·3-s + 4-s − 2.78·5-s + 2.15·6-s − 1.33·7-s − 8-s + 1.65·9-s + 2.78·10-s − 5.50·11-s − 2.15·12-s − 5.88·13-s + 1.33·14-s + 5.99·15-s + 16-s − 2.24·17-s − 1.65·18-s − 19-s − 2.78·20-s + 2.87·21-s + 5.50·22-s − 6.52·23-s + 2.15·24-s + 2.72·25-s + 5.88·26-s + 2.90·27-s − 1.33·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.24·3-s + 0.5·4-s − 1.24·5-s + 0.880·6-s − 0.503·7-s − 0.353·8-s + 0.551·9-s + 0.879·10-s − 1.65·11-s − 0.622·12-s − 1.63·13-s + 0.356·14-s + 1.54·15-s + 0.250·16-s − 0.543·17-s − 0.389·18-s − 0.229·19-s − 0.621·20-s + 0.627·21-s + 1.17·22-s − 1.36·23-s + 0.440·24-s + 0.545·25-s + 1.15·26-s + 0.558·27-s − 0.251·28-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 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 + 5.88T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 - 7.17T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 - 0.00121T + 37T^{2} \) |
| 41 | \( 1 + 0.0230T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 1.06T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 2.45T + 61T^{2} \) |
| 67 | \( 1 - 6.44T + 67T^{2} \) |
| 71 | \( 1 - 2.95T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 - 4.54T + 79T^{2} \) |
| 83 | \( 1 + 2.00T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 3.39T + 97T^{2} \) |
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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
−7.50409371430287434351838281778, −6.84782256869928230503668434026, −6.30043280869611950417423001474, −5.29263059472660488876956031810, −4.89366418382416646258496518672, −4.02914725883962212928013635556, −2.90116352916294722876575106357, −2.26485844172949572618085364352, −0.53864246789753665518813964695, 0,
0.53864246789753665518813964695, 2.26485844172949572618085364352, 2.90116352916294722876575106357, 4.02914725883962212928013635556, 4.89366418382416646258496518672, 5.29263059472660488876956031810, 6.30043280869611950417423001474, 6.84782256869928230503668434026, 7.50409371430287434351838281778
