| L(s) = 1 | + 2.59·2-s + 1.89·3-s + 4.74·4-s − 5-s + 4.91·6-s − 1.41·7-s + 7.14·8-s + 0.580·9-s − 2.59·10-s + 11-s + 8.98·12-s + 4.04·13-s − 3.68·14-s − 1.89·15-s + 9.06·16-s + 1.41·17-s + 1.50·18-s + 1.38·19-s − 4.74·20-s − 2.68·21-s + 2.59·22-s + 4.15·23-s + 13.5·24-s + 25-s + 10.5·26-s − 4.57·27-s − 6.72·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 1.09·3-s + 2.37·4-s − 0.447·5-s + 2.00·6-s − 0.535·7-s + 2.52·8-s + 0.193·9-s − 0.821·10-s + 0.301·11-s + 2.59·12-s + 1.12·13-s − 0.983·14-s − 0.488·15-s + 2.26·16-s + 0.342·17-s + 0.355·18-s + 0.318·19-s − 1.06·20-s − 0.584·21-s + 0.553·22-s + 0.866·23-s + 2.75·24-s + 0.200·25-s + 2.05·26-s − 0.880·27-s − 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.561926652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.561926652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 3 | \( 1 - 1.89T + 3T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 4.91T + 37T^{2} \) |
| 41 | \( 1 - 0.367T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 - 7.57T + 53T^{2} \) |
| 59 | \( 1 - 3.45T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 - 0.106T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 79 | \( 1 + 8.00T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.367232941267767797607331597079, −7.49528247748893890070097565629, −6.81917868621013584887678116757, −6.13045894245209003125889348923, −5.37526206251032717383256016147, −4.44480794386986526204830095133, −3.68959860825397814908211215862, −3.22067709109578257028186027253, −2.61017248073228770655127806128, −1.38997956059766905516202142606,
1.38997956059766905516202142606, 2.61017248073228770655127806128, 3.22067709109578257028186027253, 3.68959860825397814908211215862, 4.44480794386986526204830095133, 5.37526206251032717383256016147, 6.13045894245209003125889348923, 6.81917868621013584887678116757, 7.49528247748893890070097565629, 8.367232941267767797607331597079
