| L(s) = 1 | + 2-s + 4-s − 1.76·5-s + 2.53·7-s + 8-s − 1.76·10-s − 0.172·11-s + 1.33·13-s + 2.53·14-s + 16-s + 6.35·17-s + 0.244·19-s − 1.76·20-s − 0.172·22-s + 0.777·23-s − 1.87·25-s + 1.33·26-s + 2.53·28-s − 0.922·29-s − 2.24·31-s + 32-s + 6.35·34-s − 4.47·35-s + 4.59·37-s + 0.244·38-s − 1.76·40-s + 7.99·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.790·5-s + 0.956·7-s + 0.353·8-s − 0.558·10-s − 0.0520·11-s + 0.369·13-s + 0.676·14-s + 0.250·16-s + 1.54·17-s + 0.0562·19-s − 0.395·20-s − 0.0368·22-s + 0.162·23-s − 0.375·25-s + 0.261·26-s + 0.478·28-s − 0.171·29-s − 0.403·31-s + 0.176·32-s + 1.08·34-s − 0.756·35-s + 0.756·37-s + 0.0397·38-s − 0.279·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.514160447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.514160447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 1.76T + 5T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 11 | \( 1 + 0.172T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 - 6.35T + 17T^{2} \) |
| 19 | \( 1 - 0.244T + 19T^{2} \) |
| 23 | \( 1 - 0.777T + 23T^{2} \) |
| 29 | \( 1 + 0.922T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 4.59T + 37T^{2} \) |
| 41 | \( 1 - 7.99T + 41T^{2} \) |
| 43 | \( 1 + 9.41T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 1.53T + 53T^{2} \) |
| 59 | \( 1 - 7.87T + 59T^{2} \) |
| 61 | \( 1 - 1.82T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 2.04T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 0.542T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−7.67988149706791261337509704101, −7.36085357232126890095424717331, −6.31356098225178418509592943535, −5.59723994919411067610805041922, −5.04249450237278841109980232159, −4.20068972825873079939367630316, −3.68539267666744771656760974447, −2.86406352145544703144773689584, −1.81478004097303145159321636138, −0.870621774805142521334446803767,
0.870621774805142521334446803767, 1.81478004097303145159321636138, 2.86406352145544703144773689584, 3.68539267666744771656760974447, 4.20068972825873079939367630316, 5.04249450237278841109980232159, 5.59723994919411067610805041922, 6.31356098225178418509592943535, 7.36085357232126890095424717331, 7.67988149706791261337509704101
