| L(s) = 1 | − 2-s + 2·3-s + 4-s − 3.61·5-s − 2·6-s − 7-s − 8-s + 9-s + 3.61·10-s − 6.09·11-s + 2·12-s + 4.47·13-s + 14-s − 7.23·15-s + 16-s − 0.618·17-s − 18-s + 6.47·19-s − 3.61·20-s − 2·21-s + 6.09·22-s + 9.23·23-s − 2·24-s + 8.09·25-s − 4.47·26-s − 4·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 0.5·4-s − 1.61·5-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.14·10-s − 1.83·11-s + 0.577·12-s + 1.24·13-s + 0.267·14-s − 1.86·15-s + 0.250·16-s − 0.149·17-s − 0.235·18-s + 1.48·19-s − 0.809·20-s − 0.436·21-s + 1.29·22-s + 1.92·23-s − 0.408·24-s + 1.61·25-s − 0.877·26-s − 0.769·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 11 | \( 1 + 6.09T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 9.23T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 7.85T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 7.85T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 - 9.61T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 + 2.47T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.930066931623693998083362270356, −7.30511977200979479749138008294, −6.78185052237960392730051732324, −5.49104896529269686386475415110, −4.78695435683267054928957777917, −3.48033708294858068341932577557, −3.28522440673126954400552770904, −2.59557123332893417973057669544, −1.15386168089098394648524885561, 0,
1.15386168089098394648524885561, 2.59557123332893417973057669544, 3.28522440673126954400552770904, 3.48033708294858068341932577557, 4.78695435683267054928957777917, 5.49104896529269686386475415110, 6.78185052237960392730051732324, 7.30511977200979479749138008294, 7.930066931623693998083362270356
