| L(s) = 1 | − 28.8·2-s + 204.·3-s + 317.·4-s − 5.89e3·6-s + 8.86e3·7-s + 5.59e3·8-s + 2.21e4·9-s + 3.60e4·11-s + 6.49e4·12-s + 2.85e4·13-s − 2.55e5·14-s − 3.23e5·16-s − 3.27e5·17-s − 6.38e5·18-s + 2.65e5·19-s + 1.81e6·21-s − 1.03e6·22-s + 2.42e6·23-s + 1.14e6·24-s − 8.22e5·26-s + 5.09e5·27-s + 2.81e6·28-s + 3.99e6·29-s − 6.45e6·31-s + 6.46e6·32-s + 7.38e6·33-s + 9.43e6·34-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 1.45·3-s + 0.620·4-s − 1.85·6-s + 1.39·7-s + 0.483·8-s + 1.12·9-s + 0.743·11-s + 0.904·12-s + 0.277·13-s − 1.77·14-s − 1.23·16-s − 0.950·17-s − 1.43·18-s + 0.467·19-s + 2.03·21-s − 0.946·22-s + 1.80·23-s + 0.704·24-s − 0.353·26-s + 0.184·27-s + 0.865·28-s + 1.04·29-s − 1.25·31-s + 1.08·32-s + 1.08·33-s + 1.21·34-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
| \(L(5)\) |
\(\approx\) |
\(2.876702353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.876702353\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 28.8T + 512T^{2} \) |
| 3 | \( 1 - 204.T + 1.96e4T^{2} \) |
| 7 | \( 1 - 8.86e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.60e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.27e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.42e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.99e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.45e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.15e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.20e5T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.13e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.89e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.28e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.91e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.40e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.19e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.44e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.90e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.65e8T + 7.60e17T^{2} \) |
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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
−9.544617763491040763819640604789, −9.000232792526301031261681527382, −8.396091368051360478198177294923, −7.68754502120771865286162177644, −6.84480316717184622055417133640, −4.94854705104040851643764474465, −3.97326565718226871585025494842, −2.59400727779380644495301792550, −1.65470632560460030243148934990, −0.912778303395833688043970634833,
0.912778303395833688043970634833, 1.65470632560460030243148934990, 2.59400727779380644495301792550, 3.97326565718226871585025494842, 4.94854705104040851643764474465, 6.84480316717184622055417133640, 7.68754502120771865286162177644, 8.396091368051360478198177294923, 9.000232792526301031261681527382, 9.544617763491040763819640604789
