| L(s) = 1 | + 6.59·2-s − 15.2·3-s + 11.5·4-s + 19.9·5-s − 100.·6-s + 48.3·7-s − 135.·8-s − 9.35·9-s + 131.·10-s − 616.·11-s − 175.·12-s − 329.·13-s + 318.·14-s − 304.·15-s − 1.25e3·16-s − 403.·17-s − 61.6·18-s − 853.·19-s + 229.·20-s − 738.·21-s − 4.06e3·22-s + 1.50e3·23-s + 2.06e3·24-s − 2.72e3·25-s − 2.17e3·26-s + 3.85e3·27-s + 555.·28-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 0.980·3-s + 0.359·4-s + 0.356·5-s − 1.14·6-s + 0.372·7-s − 0.746·8-s − 0.0384·9-s + 0.415·10-s − 1.53·11-s − 0.352·12-s − 0.540·13-s + 0.434·14-s − 0.349·15-s − 1.23·16-s − 0.338·17-s − 0.0448·18-s − 0.542·19-s + 0.128·20-s − 0.365·21-s − 1.79·22-s + 0.593·23-s + 0.732·24-s − 0.873·25-s − 0.630·26-s + 1.01·27-s + 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + 5.04e3T \) |
| good | 2 | \( 1 - 6.59T + 32T^{2} \) |
| 3 | \( 1 + 15.2T + 243T^{2} \) |
| 5 | \( 1 - 19.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 48.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 616.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 329.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 403.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 853.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.50e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.43e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.67e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.71e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.51e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.85e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.72e4T + 1.35e9T^{2} \) |
| 73 | \( 1 - 3.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.34e5T + 8.58e9T^{2} \) |
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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
−13.11723759805464309555189536575, −12.20296877983124413501254690373, −11.21163024853652632777059633979, −10.08239671601505218521792756191, −8.340788683405881635758801590122, −6.54332654297571256555067438882, −5.40920811333008798834677325046, −4.71364094370194921761416360465, −2.68980528851994658215377546177, 0,
2.68980528851994658215377546177, 4.71364094370194921761416360465, 5.40920811333008798834677325046, 6.54332654297571256555067438882, 8.340788683405881635758801590122, 10.08239671601505218521792756191, 11.21163024853652632777059633979, 12.20296877983124413501254690373, 13.11723759805464309555189536575
