| L(s) = 1 | − 4.68·2-s − 490.·4-s + 625·5-s + 2.40e3·7-s + 4.69e3·8-s − 2.92e3·10-s − 6.12e4·11-s + 1.27e5·13-s − 1.12e4·14-s + 2.28e5·16-s + 3.74e5·17-s + 3.51e5·19-s − 3.06e5·20-s + 2.86e5·22-s − 1.55e6·23-s + 3.90e5·25-s − 5.98e5·26-s − 1.17e6·28-s − 3.12e6·29-s + 6.54e6·31-s − 3.47e6·32-s − 1.75e6·34-s + 1.50e6·35-s − 9.25e6·37-s − 1.64e6·38-s + 2.93e6·40-s + 8.05e6·41-s + ⋯ |
| L(s) = 1 | − 0.206·2-s − 0.957·4-s + 0.447·5-s + 0.377·7-s + 0.404·8-s − 0.0925·10-s − 1.26·11-s + 1.24·13-s − 0.0781·14-s + 0.873·16-s + 1.08·17-s + 0.618·19-s − 0.428·20-s + 0.261·22-s − 1.16·23-s + 0.200·25-s − 0.256·26-s − 0.361·28-s − 0.821·29-s + 1.27·31-s − 0.585·32-s − 0.224·34-s + 0.169·35-s − 0.812·37-s − 0.127·38-s + 0.181·40-s + 0.444·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.724852850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724852850\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 4.68T + 512T^{2} \) |
| 11 | \( 1 + 6.12e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.27e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.74e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.51e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.55e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.12e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.54e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.25e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 8.05e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.58e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 8.88e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.14e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.04e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.88e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.24e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.91e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.39e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.77e8T + 7.60e17T^{2} \) |
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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
−10.11681359706931137898124665035, −9.167752852212856891568976168669, −8.202470145604091148337384000018, −7.61856396061878984129642220851, −5.93303889426378193938195701163, −5.32428738747807592715270538774, −4.18771062113447505601598990517, −3.06990573541549945708675248060, −1.61870746603028050587192357780, −0.62783666133123237192773690870,
0.62783666133123237192773690870, 1.61870746603028050587192357780, 3.06990573541549945708675248060, 4.18771062113447505601598990517, 5.32428738747807592715270538774, 5.93303889426378193938195701163, 7.61856396061878984129642220851, 8.202470145604091148337384000018, 9.167752852212856891568976168669, 10.11681359706931137898124665035
