| L(s) = 1 | − 32.6·2-s − 729·3-s − 7.12e3·4-s − 4.93e4·5-s + 2.37e4·6-s − 5.93e5·7-s + 4.99e5·8-s + 5.31e5·9-s + 1.61e6·10-s − 7.46e6·11-s + 5.19e6·12-s − 3.21e7·13-s + 1.93e7·14-s + 3.59e7·15-s + 4.20e7·16-s − 1.18e7·17-s − 1.73e7·18-s − 3.23e8·19-s + 3.52e8·20-s + 4.32e8·21-s + 2.43e8·22-s − 7.23e8·23-s − 3.64e8·24-s + 1.21e9·25-s + 1.04e9·26-s − 3.87e8·27-s + 4.22e9·28-s + ⋯ |
| L(s) = 1 | − 0.360·2-s − 0.577·3-s − 0.870·4-s − 1.41·5-s + 0.208·6-s − 1.90·7-s + 0.673·8-s + 0.333·9-s + 0.509·10-s − 1.27·11-s + 0.502·12-s − 1.84·13-s + 0.686·14-s + 0.816·15-s + 0.627·16-s − 0.119·17-s − 0.120·18-s − 1.57·19-s + 1.22·20-s + 1.10·21-s + 0.457·22-s − 1.01·23-s − 0.389·24-s + 0.997·25-s + 0.665·26-s − 0.192·27-s + 1.65·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 729T \) |
| 59 | \( 1 + 4.21e10T \) |
| good | 2 | \( 1 + 32.6T + 8.19e3T^{2} \) |
| 5 | \( 1 + 4.93e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.93e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 7.46e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 3.21e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.18e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.23e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 7.23e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.75e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 3.99e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.32e7T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.57e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 5.80e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 6.92e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 3.07e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 1.31e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 2.68e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.88e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 7.87e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.97e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.48e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + 5.76e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.37e12T + 6.73e25T^{2} \) |
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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
−10.01467061618289374332554490553, −8.910826940334968784803514780173, −7.74279695606033976626879282901, −7.07040287386284599645018330262, −5.69251641420637026112796825400, −4.52120107747954841842359387045, −3.73877138961289353518056263492, −2.49550349787349474346430479494, −0.26144212581851694506302727379, 0,
0.26144212581851694506302727379, 2.49550349787349474346430479494, 3.73877138961289353518056263492, 4.52120107747954841842359387045, 5.69251641420637026112796825400, 7.07040287386284599645018330262, 7.74279695606033976626879282901, 8.910826940334968784803514780173, 10.01467061618289374332554490553
