| L(s) = 1 | + 1.25·2-s − 6.43·4-s + 5.67·7-s − 18.0·8-s − 21.2·11-s − 40.9·13-s + 7.09·14-s + 28.8·16-s + 59.1·17-s + 21.8·19-s − 26.5·22-s − 98.7·23-s − 51.2·26-s − 36.4·28-s + 159.·29-s − 69.4·31-s + 180.·32-s + 74.0·34-s + 235.·37-s + 27.3·38-s + 491.·41-s + 95.3·43-s + 136.·44-s − 123.·46-s − 548.·47-s − 310.·49-s + 263.·52-s + ⋯ |
| L(s) = 1 | + 0.442·2-s − 0.804·4-s + 0.306·7-s − 0.798·8-s − 0.582·11-s − 0.873·13-s + 0.135·14-s + 0.451·16-s + 0.844·17-s + 0.264·19-s − 0.257·22-s − 0.895·23-s − 0.386·26-s − 0.246·28-s + 1.02·29-s − 0.402·31-s + 0.997·32-s + 0.373·34-s + 1.04·37-s + 0.116·38-s + 1.87·41-s + 0.338·43-s + 0.468·44-s − 0.396·46-s − 1.70·47-s − 0.906·49-s + 0.702·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.683462111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683462111\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.25T + 8T^{2} \) |
| 7 | \( 1 - 5.67T + 343T^{2} \) |
| 11 | \( 1 + 21.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 98.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 69.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 235.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 491.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 95.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 548.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 509.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 741.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 387.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 508.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 914.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 925.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 708.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.45e3T + 9.12e5T^{2} \) |
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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
−9.890212196841404743095242827386, −9.451366789595046859162398839666, −8.183133899766903843461166459828, −7.72526378294249314675198601141, −6.33193236188468574747189429355, −5.34726847478434689860375454594, −4.68858555581077858333915699658, −3.61530032560255252320590118974, −2.45231743918533493121436469465, −0.70296695503869248308554323077,
0.70296695503869248308554323077, 2.45231743918533493121436469465, 3.61530032560255252320590118974, 4.68858555581077858333915699658, 5.34726847478434689860375454594, 6.33193236188468574747189429355, 7.72526378294249314675198601141, 8.183133899766903843461166459828, 9.451366789595046859162398839666, 9.890212196841404743095242827386
