L(s) = 1 | − 0.697·2-s + 1.52·3-s − 1.51·4-s + 5-s − 1.06·6-s + 7-s + 2.44·8-s − 0.662·9-s − 0.697·10-s + 4.72·11-s − 2.31·12-s + 3.96·13-s − 0.697·14-s + 1.52·15-s + 1.31·16-s − 7.67·17-s + 0.461·18-s + 3.77·19-s − 1.51·20-s + 1.52·21-s − 3.29·22-s − 4.65·23-s + 3.74·24-s + 25-s − 2.76·26-s − 5.59·27-s − 1.51·28-s + ⋯ |
L(s) = 1 | − 0.492·2-s + 0.882·3-s − 0.756·4-s + 0.447·5-s − 0.435·6-s + 0.377·7-s + 0.866·8-s − 0.220·9-s − 0.220·10-s + 1.42·11-s − 0.668·12-s + 1.09·13-s − 0.186·14-s + 0.394·15-s + 0.329·16-s − 1.86·17-s + 0.108·18-s + 0.866·19-s − 0.338·20-s + 0.333·21-s − 0.702·22-s − 0.969·23-s + 0.764·24-s + 0.200·25-s − 0.541·26-s − 1.07·27-s − 0.286·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 + 0.697T + 2T^{2} \) |
| 3 | \( 1 - 1.52T + 3T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 13 | \( 1 - 3.96T + 13T^{2} \) |
| 17 | \( 1 + 7.67T + 17T^{2} \) |
| 19 | \( 1 - 3.77T + 19T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 + 9.27T + 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 + 8.01T + 37T^{2} \) |
| 41 | \( 1 + 3.28T + 41T^{2} \) |
| 43 | \( 1 - 5.79T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 - 0.616T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 6.08T + 71T^{2} \) |
| 73 | \( 1 + 4.50T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 7.55T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−7.57963489009400749848789640214, −7.06932027872818899311086363492, −5.99137278017812894954356224832, −5.56072667198460321336495314704, −4.33454921314767906480276856212, −3.97607952075419103760824229932, −3.18800357166173700023659305137, −1.89010838760189718385749627226, −1.50891471532227974248845563848, 0,
1.50891471532227974248845563848, 1.89010838760189718385749627226, 3.18800357166173700023659305137, 3.97607952075419103760824229932, 4.33454921314767906480276856212, 5.56072667198460321336495314704, 5.99137278017812894954356224832, 7.06932027872818899311086363492, 7.57963489009400749848789640214