| L(s) = 1 | − 263.·3-s + 2.08e3·5-s − 3.94e3·7-s + 4.99e4·9-s + 3.27e4·11-s + 2.85e4·13-s − 5.49e5·15-s + 5.79e5·17-s + 9.09e5·19-s + 1.04e6·21-s + 1.86e6·23-s + 2.38e6·25-s − 7.98e6·27-s + 4.60e6·29-s − 2.51e6·31-s − 8.64e6·33-s − 8.23e6·35-s − 1.29e7·37-s − 7.53e6·39-s − 3.17e7·41-s + 2.64e7·43-s + 1.04e8·45-s + 2.69e7·47-s − 2.47e7·49-s − 1.52e8·51-s + 1.30e7·53-s + 6.82e7·55-s + ⋯ |
| L(s) = 1 | − 1.88·3-s + 1.49·5-s − 0.621·7-s + 2.53·9-s + 0.674·11-s + 0.277·13-s − 2.80·15-s + 1.68·17-s + 1.60·19-s + 1.16·21-s + 1.39·23-s + 1.22·25-s − 2.89·27-s + 1.20·29-s − 0.489·31-s − 1.26·33-s − 0.927·35-s − 1.13·37-s − 0.521·39-s − 1.75·41-s + 1.18·43-s + 3.78·45-s + 0.805·47-s − 0.613·49-s − 3.16·51-s + 0.226·53-s + 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.946500805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.946500805\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 + 263.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.08e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.94e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.27e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 5.79e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.09e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.86e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.60e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.51e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.29e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.17e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.64e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.69e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.30e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.91e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.36e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.54e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.56e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.79e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.94e6T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.96e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.74e7T + 7.60e17T^{2} \) |
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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
−10.61300411708570977667248015374, −9.982583152261371312101690013048, −9.266993653181199883891093954798, −7.20312130987559810680431268576, −6.42223661585254681805255653435, −5.59992839290322651016119038290, −5.06109431947047561014056499975, −3.32140316424767598711303706214, −1.42763050475613827368689991881, −0.850704310360320515060700880174,
0.850704310360320515060700880174, 1.42763050475613827368689991881, 3.32140316424767598711303706214, 5.06109431947047561014056499975, 5.59992839290322651016119038290, 6.42223661585254681805255653435, 7.20312130987559810680431268576, 9.266993653181199883891093954798, 9.982583152261371312101690013048, 10.61300411708570977667248015374
