| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 21.3·5-s + 6·6-s + 12.1·7-s − 8·8-s + 9·9-s − 42.7·10-s − 2.11·11-s − 12·12-s − 63.2·13-s − 24.3·14-s − 64.1·15-s + 16·16-s − 89.3·17-s − 18·18-s + 85.5·20-s − 36.5·21-s + 4.23·22-s + 75.0·23-s + 24·24-s + 332.·25-s + 126.·26-s − 27·27-s + 48.6·28-s + 39.0·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.91·5-s + 0.408·6-s + 0.657·7-s − 0.353·8-s + 0.333·9-s − 1.35·10-s − 0.0580·11-s − 0.288·12-s − 1.35·13-s − 0.464·14-s − 1.10·15-s + 0.250·16-s − 1.27·17-s − 0.235·18-s + 0.956·20-s − 0.379·21-s + 0.0410·22-s + 0.680·23-s + 0.204·24-s + 2.66·25-s + 0.954·26-s − 0.192·27-s + 0.328·28-s + 0.249·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 21.3T + 125T^{2} \) |
| 7 | \( 1 - 12.1T + 343T^{2} \) |
| 11 | \( 1 + 2.11T + 1.33e3T^{2} \) |
| 13 | \( 1 + 63.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 89.3T + 4.91e3T^{2} \) |
| 23 | \( 1 - 75.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 39.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 15.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 98.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 37.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 387.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 294.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 126.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 572.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 734.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 829.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 444.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 726.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 117.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−8.498070540189782728657915056335, −7.49209870991265306966679572595, −6.61714302753247571481004953671, −6.19317802910857810349177004000, −5.03618374963479505891532660312, −4.82778797773730357963784696590, −2.86821768806920690403847096727, −2.03912384153354343419409209018, −1.38195741296889355977441665171, 0,
1.38195741296889355977441665171, 2.03912384153354343419409209018, 2.86821768806920690403847096727, 4.82778797773730357963784696590, 5.03618374963479505891532660312, 6.19317802910857810349177004000, 6.61714302753247571481004953671, 7.49209870991265306966679572595, 8.498070540189782728657915056335
