| L(s) = 1 | + 3.97·2-s + 3·3-s + 7.77·4-s + 9.11·5-s + 11.9·6-s − 24.3·7-s − 0.887·8-s + 9·9-s + 36.2·10-s − 11·11-s + 23.3·12-s + 20.0·13-s − 96.6·14-s + 27.3·15-s − 65.7·16-s + 25.0·17-s + 35.7·18-s − 66.8·19-s + 70.8·20-s − 72.9·21-s − 43.6·22-s + 35.3·23-s − 2.66·24-s − 41.9·25-s + 79.7·26-s + 27·27-s − 189.·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.577·3-s + 0.972·4-s + 0.815·5-s + 0.810·6-s − 1.31·7-s − 0.0392·8-s + 0.333·9-s + 1.14·10-s − 0.301·11-s + 0.561·12-s + 0.428·13-s − 1.84·14-s + 0.470·15-s − 1.02·16-s + 0.357·17-s + 0.468·18-s − 0.807·19-s + 0.792·20-s − 0.758·21-s − 0.423·22-s + 0.320·23-s − 0.0226·24-s − 0.335·25-s + 0.601·26-s + 0.192·27-s − 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 3.97T + 8T^{2} \) |
| 5 | \( 1 - 9.11T + 125T^{2} \) |
| 7 | \( 1 + 24.3T + 343T^{2} \) |
| 13 | \( 1 - 20.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 66.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 78.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 67.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 174.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 228.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 165.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 408.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 42.9T + 2.05e5T^{2} \) |
| 67 | \( 1 + 578.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 680.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 257.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 988.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 990.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 964.T + 9.12e5T^{2} \) |
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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
−8.571396267950742867362031421914, −7.28070793024181026245155966363, −6.55083687488674434988436741868, −5.91239281052626253257033628199, −5.25347917376198202672300785927, −4.13652055495101591631521169661, −3.42447250796687215353327951461, −2.74296989484462906054874423203, −1.79002619125484448520229819682, 0,
1.79002619125484448520229819682, 2.74296989484462906054874423203, 3.42447250796687215353327951461, 4.13652055495101591631521169661, 5.25347917376198202672300785927, 5.91239281052626253257033628199, 6.55083687488674434988436741868, 7.28070793024181026245155966363, 8.571396267950742867362031421914
