| L(s) = 1 | − 23.7·2-s − 595.·3-s − 1.48e3·4-s − 5.29e3·5-s + 1.41e4·6-s + 8.04e4·7-s + 8.39e4·8-s + 1.77e5·9-s + 1.25e5·10-s − 4.88e4·11-s + 8.83e5·12-s − 9.30e5·13-s − 1.91e6·14-s + 3.15e6·15-s + 1.04e6·16-s + 5.06e5·17-s − 4.22e6·18-s + 1.04e6·19-s + 7.85e6·20-s − 4.79e7·21-s + 1.16e6·22-s + 2.83e7·23-s − 4.99e7·24-s − 2.07e7·25-s + 2.21e7·26-s − 3.96e5·27-s − 1.19e8·28-s + ⋯ |
| L(s) = 1 | − 0.525·2-s − 1.41·3-s − 0.724·4-s − 0.758·5-s + 0.743·6-s + 1.81·7-s + 0.905·8-s + 1.00·9-s + 0.398·10-s − 0.0914·11-s + 1.02·12-s − 0.694·13-s − 0.950·14-s + 1.07·15-s + 0.248·16-s + 0.0864·17-s − 0.527·18-s + 0.0970·19-s + 0.549·20-s − 2.56·21-s + 0.0480·22-s + 0.918·23-s − 1.28·24-s − 0.425·25-s + 0.364·26-s − 0.00532·27-s − 1.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 2.05e7T \) |
| good | 2 | \( 1 + 23.7T + 2.04e3T^{2} \) |
| 3 | \( 1 + 595.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 5.29e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 8.04e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 4.88e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 9.30e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.06e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.04e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.83e7T + 9.52e14T^{2} \) |
| 31 | \( 1 - 1.94e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 7.56e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.07e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.48e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.66e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 8.34e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 6.60e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 7.10e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 8.27e8T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.25e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 6.51e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.48e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.41e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.45e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.19e11T + 7.15e21T^{2} \) |
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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
−14.00509493277143192855453859395, −12.21779297592096224835798554943, −11.36207283962560414296573496211, −10.35845752513506765030161085875, −8.503831802455798364497623720562, −7.39645537136669881449694195081, −5.23552079902520007299860888186, −4.49827824122112019126226150835, −1.24281806890088297027605842609, 0,
1.24281806890088297027605842609, 4.49827824122112019126226150835, 5.23552079902520007299860888186, 7.39645537136669881449694195081, 8.503831802455798364497623720562, 10.35845752513506765030161085875, 11.36207283962560414296573496211, 12.21779297592096224835798554943, 14.00509493277143192855453859395
