| L(s) = 1 | + 38.3·2-s − 137.·3-s + 958.·4-s + 2.13e3·5-s − 5.28e3·6-s − 2.99e3·7-s + 1.71e4·8-s − 695.·9-s + 8.17e4·10-s + 7.54e4·11-s − 1.32e5·12-s + 7.86e4·13-s − 1.14e5·14-s − 2.93e5·15-s + 1.65e5·16-s + 2.91e5·17-s − 2.66e4·18-s − 6.06e4·19-s + 2.04e6·20-s + 4.12e5·21-s + 2.89e6·22-s + 3.05e5·23-s − 2.35e6·24-s + 2.58e6·25-s + 3.01e6·26-s + 2.80e6·27-s − 2.86e6·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s − 0.982·3-s + 1.87·4-s + 1.52·5-s − 1.66·6-s − 0.471·7-s + 1.47·8-s − 0.0353·9-s + 2.58·10-s + 1.55·11-s − 1.83·12-s + 0.763·13-s − 0.798·14-s − 1.49·15-s + 0.632·16-s + 0.847·17-s − 0.0598·18-s − 0.106·19-s + 2.85·20-s + 0.462·21-s + 2.63·22-s + 0.227·23-s − 1.45·24-s + 1.32·25-s + 1.29·26-s + 1.01·27-s − 0.882·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.912083871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.912083871\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 38.3T + 512T^{2} \) |
| 3 | \( 1 + 137.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.13e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.99e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.54e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.86e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.91e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.06e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.05e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.63e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.26e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.91e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 5.35e5T + 3.27e14T^{2} \) |
| 47 | \( 1 + 6.22e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.94e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.92e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.45e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.98e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.32e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.87e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 9.46e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.93e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.10e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.41e9T + 7.60e17T^{2} \) |
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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
−13.93829958756108266585302345137, −12.88572298759337913063401347898, −11.92763858772702211170832651826, −10.84313192250154666465486107720, −9.370689319752499798244429452596, −6.37564533053301081539169481788, −6.18234267216998966044163516324, −4.96026621334949528444194547584, −3.28313331698807635425015200475, −1.48451118967508721242706839304,
1.48451118967508721242706839304, 3.28313331698807635425015200475, 4.96026621334949528444194547584, 6.18234267216998966044163516324, 6.37564533053301081539169481788, 9.370689319752499798244429452596, 10.84313192250154666465486107720, 11.92763858772702211170832651826, 12.88572298759337913063401347898, 13.93829958756108266585302345137
