| L(s) = 1 | − 1.17·2-s + 2.35·3-s − 0.625·4-s − 4.07·5-s − 2.75·6-s − 0.390·7-s + 3.07·8-s + 2.53·9-s + 4.77·10-s + 5.79·11-s − 1.47·12-s − 13-s + 0.457·14-s − 9.58·15-s − 2.35·16-s − 5.92·17-s − 2.97·18-s − 0.102·19-s + 2.54·20-s − 0.919·21-s − 6.79·22-s − 0.876·23-s + 7.24·24-s + 11.6·25-s + 1.17·26-s − 1.08·27-s + 0.244·28-s + ⋯ |
| L(s) = 1 | − 0.829·2-s + 1.35·3-s − 0.312·4-s − 1.82·5-s − 1.12·6-s − 0.147·7-s + 1.08·8-s + 0.845·9-s + 1.51·10-s + 1.74·11-s − 0.424·12-s − 0.277·13-s + 0.122·14-s − 2.47·15-s − 0.589·16-s − 1.43·17-s − 0.701·18-s − 0.0235·19-s + 0.569·20-s − 0.200·21-s − 1.44·22-s − 0.182·23-s + 1.47·24-s + 2.32·25-s + 0.229·26-s − 0.209·27-s + 0.0461·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 + 4.07T + 5T^{2} \) |
| 7 | \( 1 + 0.390T + 7T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 17 | \( 1 + 5.92T + 17T^{2} \) |
| 19 | \( 1 + 0.102T + 19T^{2} \) |
| 23 | \( 1 + 0.876T + 23T^{2} \) |
| 29 | \( 1 + 9.55T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 + 4.79T + 37T^{2} \) |
| 41 | \( 1 - 0.995T + 41T^{2} \) |
| 43 | \( 1 + 7.84T + 43T^{2} \) |
| 47 | \( 1 + 4.91T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 - 7.00T + 61T^{2} \) |
| 67 | \( 1 + 7.41T + 67T^{2} \) |
| 71 | \( 1 + 1.18T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.152779386659337560068968298441, −8.714239422704941934611501297256, −8.143254356316248764183410822372, −7.36715924710355606711362741132, −6.69864917701758640825331975082, −4.62634254938884658099665034542, −3.98651446793426961216469649459, −3.33147650189770563273396869054, −1.70616988216174945669545147858, 0,
1.70616988216174945669545147858, 3.33147650189770563273396869054, 3.98651446793426961216469649459, 4.62634254938884658099665034542, 6.69864917701758640825331975082, 7.36715924710355606711362741132, 8.143254356316248764183410822372, 8.714239422704941934611501297256, 9.152779386659337560068968298441
