| L(s) = 1 | + 1.69·2-s + 0.868·4-s − 3.51·5-s − 7-s − 1.91·8-s − 5.94·10-s + 1.74·11-s + 6.33·13-s − 1.69·14-s − 4.98·16-s + 5.94·17-s + 1.74·19-s − 3.04·20-s + 2.95·22-s − 23-s + 7.33·25-s + 10.7·26-s − 0.868·28-s + 5.68·29-s + 7.94·31-s − 4.60·32-s + 10.0·34-s + 3.51·35-s + 1.53·37-s + 2.95·38-s + 6.73·40-s − 12.1·41-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 0.434·4-s − 1.57·5-s − 0.377·7-s − 0.677·8-s − 1.88·10-s + 0.525·11-s + 1.75·13-s − 0.452·14-s − 1.24·16-s + 1.44·17-s + 0.399·19-s − 0.681·20-s + 0.629·22-s − 0.208·23-s + 1.46·25-s + 2.10·26-s − 0.164·28-s + 1.05·29-s + 1.42·31-s − 0.813·32-s + 1.72·34-s + 0.593·35-s + 0.252·37-s + 0.478·38-s + 1.06·40-s − 1.89·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.245318506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.245318506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 5 | \( 1 + 3.51T + 5T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 - 6.33T + 13T^{2} \) |
| 17 | \( 1 - 5.94T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 - 7.94T + 31T^{2} \) |
| 37 | \( 1 - 1.53T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 3.59T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 8.69T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 + 4.46T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 + 7.42T + 79T^{2} \) |
| 83 | \( 1 - 7.75T + 83T^{2} \) |
| 89 | \( 1 + 4.18T + 89T^{2} \) |
| 97 | \( 1 - 5.53T + 97T^{2} \) |
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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
−9.490220715302552333865289955878, −8.421049108402573100020484642065, −8.053788204894699198274197859150, −6.79706706781244271811737381784, −6.21459065373180500282680225349, −5.19096427706658645062863988854, −4.24508431567084590996316295353, −3.57290856023817880992475577789, −3.11998907675590943133095924404, −0.935061112879754116572941088963,
0.935061112879754116572941088963, 3.11998907675590943133095924404, 3.57290856023817880992475577789, 4.24508431567084590996316295353, 5.19096427706658645062863988854, 6.21459065373180500282680225349, 6.79706706781244271811737381784, 8.053788204894699198274197859150, 8.421049108402573100020484642065, 9.490220715302552333865289955878
