| L(s) = 1 | + 22.9·2-s + 40.8·3-s + 16.4·4-s + 938.·6-s − 7.84e3·7-s − 1.13e4·8-s − 1.80e4·9-s − 3.70e4·11-s + 671.·12-s − 2.85e4·13-s − 1.80e5·14-s − 2.70e5·16-s − 6.40e4·17-s − 4.14e5·18-s + 7.65e5·19-s − 3.20e5·21-s − 8.51e5·22-s − 1.32e6·23-s − 4.65e5·24-s − 6.56e5·26-s − 1.53e6·27-s − 1.28e5·28-s − 1.64e6·29-s − 1.02e7·31-s − 3.80e5·32-s − 1.51e6·33-s − 1.47e6·34-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.291·3-s + 0.0320·4-s + 0.295·6-s − 1.23·7-s − 0.983·8-s − 0.915·9-s − 0.762·11-s + 0.00934·12-s − 0.277·13-s − 1.25·14-s − 1.03·16-s − 0.186·17-s − 0.929·18-s + 1.34·19-s − 0.359·21-s − 0.775·22-s − 0.987·23-s − 0.286·24-s − 0.281·26-s − 0.557·27-s − 0.0396·28-s − 0.431·29-s − 1.99·31-s − 0.0641·32-s − 0.222·33-s − 0.189·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.095394257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.095394257\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 22.9T + 512T^{2} \) |
| 3 | \( 1 - 40.8T + 1.96e4T^{2} \) |
| 7 | \( 1 + 7.84e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.70e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 6.40e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.32e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.64e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.02e7T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.52e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.66e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.88e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.25e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.46e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.46e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.86e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.77e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.22e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.64e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.86e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.75e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.84e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.12e9T + 7.60e17T^{2} \) |
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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
−9.838653514939658115437284556840, −9.286375945674327553212097018340, −8.177561175622015134229438657984, −7.01765522701241973816230710678, −5.81356838535887003737920828286, −5.36046104198269596841158314164, −3.92770142664422647034961970068, −3.18139358840448787234034434035, −2.38765935426073555769075617857, −0.35474746723814180214994732233,
0.35474746723814180214994732233, 2.38765935426073555769075617857, 3.18139358840448787234034434035, 3.92770142664422647034961970068, 5.36046104198269596841158314164, 5.81356838535887003737920828286, 7.01765522701241973816230710678, 8.177561175622015134229438657984, 9.286375945674327553212097018340, 9.838653514939658115437284556840
