| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 4·7-s − 8-s + 9-s + 10-s − 5·11-s − 12-s − 4·14-s + 15-s + 16-s + 17-s − 18-s + 7·19-s − 20-s − 4·21-s + 5·22-s − 6·23-s + 24-s + 25-s − 27-s + 4·28-s + 4·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s − 0.288·12-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.60·19-s − 0.223·20-s − 0.872·21-s + 1.06·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.755·28-s + 0.742·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.860582160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.860582160\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.88425513024961, −13.66613219926665, −12.63169954996523, −12.25775751110294, −11.82498855769440, −11.43082507761987, −10.88877195343146, −10.50341093662865, −10.00878410132908, −9.580368113438849, −8.698550552841690, −8.216184101434006, −7.828395747425331, −7.607211927832830, −6.922229604505701, −6.249967574681280, −5.456258831849504, −5.230069343948136, −4.693411719330773, −3.985344288221586, −3.225747437431266, −2.415866497576685, −1.987978901839260, −0.9295537464950858, −0.6714984493885454,
0.6714984493885454, 0.9295537464950858, 1.987978901839260, 2.415866497576685, 3.225747437431266, 3.985344288221586, 4.693411719330773, 5.230069343948136, 5.456258831849504, 6.249967574681280, 6.922229604505701, 7.607211927832830, 7.828395747425331, 8.216184101434006, 8.698550552841690, 9.580368113438849, 10.00878410132908, 10.50341093662865, 10.88877195343146, 11.43082507761987, 11.82498855769440, 12.25775751110294, 12.63169954996523, 13.66613219926665, 13.88425513024961
