| L(s) = 1 | + 81·3-s + 1.65e3·5-s + 1.13e4·7-s + 6.56e3·9-s − 4.38e4·11-s − 2.01e4·13-s + 1.34e5·15-s + 1.07e5·17-s + 8.70e5·19-s + 9.20e5·21-s − 1.61e6·23-s + 7.95e5·25-s + 5.31e5·27-s − 7.11e5·29-s − 4.84e6·31-s − 3.54e6·33-s + 1.88e7·35-s + 3.54e5·37-s − 1.63e6·39-s + 2.36e7·41-s + 5.28e6·43-s + 1.08e7·45-s + 5.51e7·47-s + 8.89e7·49-s + 8.73e6·51-s + 5.31e7·53-s − 7.26e7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.18·5-s + 1.78·7-s + 0.333·9-s − 0.902·11-s − 0.196·13-s + 0.684·15-s + 0.313·17-s + 1.53·19-s + 1.03·21-s − 1.20·23-s + 0.407·25-s + 0.192·27-s − 0.186·29-s − 0.941·31-s − 0.521·33-s + 2.12·35-s + 0.0310·37-s − 0.113·39-s + 1.30·41-s + 0.235·43-s + 0.395·45-s + 1.64·47-s + 2.20·49-s + 0.180·51-s + 0.925·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.126674778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.126674778\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 81T \) |
| good | 5 | \( 1 - 1.65e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.13e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.38e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.01e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.07e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.70e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.61e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.11e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.84e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.54e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.36e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 5.28e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.51e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.31e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.35e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.20e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.20e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.81e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.45e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.34e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.29e9T + 7.60e17T^{2} \) |
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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
−9.805802721183779733689833991647, −8.913758881128093378621751296589, −7.88573077087244520348870993756, −7.39630235047719878961851749492, −5.67890540998574231513375271654, −5.26486742461106083882064829493, −4.05322848445806935175790153026, −2.55055231197873506217579348191, −1.91689447585586066955072201677, −0.975492411677741652025186250866,
0.975492411677741652025186250866, 1.91689447585586066955072201677, 2.55055231197873506217579348191, 4.05322848445806935175790153026, 5.26486742461106083882064829493, 5.67890540998574231513375271654, 7.39630235047719878961851749492, 7.88573077087244520348870993756, 8.913758881128093378621751296589, 9.805802721183779733689833991647
