| L(s) = 1 | − 2.64·2-s + 4.98·4-s − 5-s + 7-s − 7.87·8-s + 2.64·10-s − 1.63·11-s − 0.915·13-s − 2.64·14-s + 10.8·16-s + 7.74·17-s + 2.19·19-s − 4.98·20-s + 4.30·22-s + 23-s + 25-s + 2.41·26-s + 4.98·28-s + 9.95·29-s + 0.201·31-s − 12.9·32-s − 20.4·34-s − 35-s + 5.93·37-s − 5.80·38-s + 7.87·40-s + 12.4·41-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 2.49·4-s − 0.447·5-s + 0.377·7-s − 2.78·8-s + 0.835·10-s − 0.491·11-s − 0.253·13-s − 0.706·14-s + 2.71·16-s + 1.87·17-s + 0.504·19-s − 1.11·20-s + 0.918·22-s + 0.208·23-s + 0.200·25-s + 0.474·26-s + 0.941·28-s + 1.84·29-s + 0.0362·31-s − 2.28·32-s − 3.51·34-s − 0.169·35-s + 0.975·37-s − 0.942·38-s + 1.24·40-s + 1.94·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8908735312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8908735312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 13 | \( 1 + 0.915T + 13T^{2} \) |
| 17 | \( 1 - 7.74T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 29 | \( 1 - 9.95T + 29T^{2} \) |
| 31 | \( 1 - 0.201T + 31T^{2} \) |
| 37 | \( 1 - 5.93T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 + 3.72T + 53T^{2} \) |
| 59 | \( 1 + 0.385T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 1.78T + 67T^{2} \) |
| 71 | \( 1 + 4.27T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 9.80T + 79T^{2} \) |
| 83 | \( 1 + 9.33T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.059649823565488067539241577827, −7.49931142163254136034546430209, −6.96944413358373280568929762173, −6.02165127322681826217148992390, −5.36305756687633789304928186384, −4.28183470971037531770538059840, −3.04845619471339864451949195175, −2.59001794959204712796517683416, −1.31436046476808226115444145952, −0.72054799008290791667783113600,
0.72054799008290791667783113600, 1.31436046476808226115444145952, 2.59001794959204712796517683416, 3.04845619471339864451949195175, 4.28183470971037531770538059840, 5.36305756687633789304928186384, 6.02165127322681826217148992390, 6.96944413358373280568929762173, 7.49931142163254136034546430209, 8.059649823565488067539241577827
