L(s) = 1 | + 2-s − 1.73·3-s + 4-s + 3.77·5-s − 1.73·6-s + 0.621·7-s + 8-s + 0.00185·9-s + 3.77·10-s − 4.90·11-s − 1.73·12-s − 3.53·13-s + 0.621·14-s − 6.54·15-s + 16-s − 2.59·17-s + 0.00185·18-s + 5.60·19-s + 3.77·20-s − 1.07·21-s − 4.90·22-s − 3.80·23-s − 1.73·24-s + 9.26·25-s − 3.53·26-s + 5.19·27-s + 0.621·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.00·3-s + 0.5·4-s + 1.68·5-s − 0.707·6-s + 0.234·7-s + 0.353·8-s + 0.000617·9-s + 1.19·10-s − 1.48·11-s − 0.500·12-s − 0.980·13-s + 0.166·14-s − 1.68·15-s + 0.250·16-s − 0.629·17-s + 0.000436·18-s + 1.28·19-s + 0.844·20-s − 0.234·21-s − 1.04·22-s − 0.792·23-s − 0.353·24-s + 1.85·25-s − 0.693·26-s + 0.999·27-s + 0.117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 - 3.77T + 5T^{2} \) |
| 7 | \( 1 - 0.621T + 7T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 + 3.80T + 23T^{2} \) |
| 29 | \( 1 + 9.57T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 + 7.86T + 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 - 6.25T + 43T^{2} \) |
| 47 | \( 1 - 3.02T + 47T^{2} \) |
| 53 | \( 1 + 7.54T + 53T^{2} \) |
| 59 | \( 1 + 5.00T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 3.02T + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.75607476990676183538983613305, −7.20473078901842016282799621079, −6.19667945641169457478760199297, −5.74255144287318783311526029014, −5.12327140515234904771653224554, −4.86399784657292484114658646704, −3.29673804866726579414604475069, −2.38342505637838759842652177168, −1.71662671658685747635790155468, 0,
1.71662671658685747635790155468, 2.38342505637838759842652177168, 3.29673804866726579414604475069, 4.86399784657292484114658646704, 5.12327140515234904771653224554, 5.74255144287318783311526029014, 6.19667945641169457478760199297, 7.20473078901842016282799621079, 7.75607476990676183538983613305