| L(s) = 1 | − 41.4·2-s + 157.·3-s + 1.20e3·4-s + 2.66e3·5-s − 6.54e3·6-s − 2.40e3·7-s − 2.87e4·8-s + 5.24e3·9-s − 1.10e5·10-s + 2.01e4·11-s + 1.90e5·12-s + 2.85e4·13-s + 9.94e4·14-s + 4.20e5·15-s + 5.72e5·16-s − 423.·17-s − 2.17e5·18-s + 6.65e5·19-s + 3.20e6·20-s − 3.79e5·21-s − 8.35e5·22-s − 1.76e6·23-s − 4.53e6·24-s + 5.13e6·25-s − 1.18e6·26-s − 2.27e6·27-s − 2.89e6·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 1.12·3-s + 2.35·4-s + 1.90·5-s − 2.06·6-s − 0.377·7-s − 2.47·8-s + 0.266·9-s − 3.48·10-s + 0.415·11-s + 2.64·12-s + 0.277·13-s + 0.692·14-s + 2.14·15-s + 2.18·16-s − 0.00123·17-s − 0.488·18-s + 1.17·19-s + 4.48·20-s − 0.425·21-s − 0.760·22-s − 1.31·23-s − 2.78·24-s + 2.62·25-s − 0.507·26-s − 0.825·27-s − 0.889·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.986133873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.986133873\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 + 41.4T + 512T^{2} \) |
| 3 | \( 1 - 157.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.66e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 2.01e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 423.T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.76e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.48e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.88e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.30e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.67e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.23e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.00e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.07e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.39e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 4.76e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.02e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.61e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.06e6T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.32e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.99e8T + 7.60e17T^{2} \) |
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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
−12.01174940471943104530983535095, −10.37467670221870920950869961497, −9.807463191556407535542318240277, −9.060344754807672847941224541537, −8.299196203208233731690600097166, −6.88330518821939679777423793870, −5.88027918760578033495512408785, −2.94640750291478185168349499906, −2.07706803950931581639953494493, −1.05640460203827401649874995736,
1.05640460203827401649874995736, 2.07706803950931581639953494493, 2.94640750291478185168349499906, 5.88027918760578033495512408785, 6.88330518821939679777423793870, 8.299196203208233731690600097166, 9.060344754807672847941224541537, 9.807463191556407535542318240277, 10.37467670221870920950869961497, 12.01174940471943104530983535095
