| L(s) = 1 | − 4·2-s − 25.5·3-s + 16·4-s − 14·5-s + 102.·6-s + 203.·7-s − 64·8-s + 408.·9-s + 56·10-s + 60.7·11-s − 408.·12-s − 726.·13-s − 812.·14-s + 357.·15-s + 256·16-s + 1.66e3·17-s − 1.63e3·18-s − 2.10e3·19-s − 224·20-s − 5.18e3·21-s − 243.·22-s − 518.·23-s + 1.63e3·24-s − 2.92e3·25-s + 2.90e3·26-s − 4.22e3·27-s + 3.24e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.63·3-s + 0.5·4-s − 0.250·5-s + 1.15·6-s + 1.56·7-s − 0.353·8-s + 1.68·9-s + 0.177·10-s + 0.151·11-s − 0.818·12-s − 1.19·13-s − 1.10·14-s + 0.410·15-s + 0.250·16-s + 1.39·17-s − 1.18·18-s − 1.33·19-s − 0.125·20-s − 2.56·21-s − 0.107·22-s − 0.204·23-s + 0.578·24-s − 0.937·25-s + 0.842·26-s − 1.11·27-s + 0.783·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 4T \) |
| 37 | \( 1 + 1.36e3T \) |
| good | 3 | \( 1 + 25.5T + 243T^{2} \) |
| 5 | \( 1 + 14T + 3.12e3T^{2} \) |
| 7 | \( 1 - 203.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 60.7T + 1.61e5T^{2} \) |
| 13 | \( 1 + 726.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.66e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 518.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.04e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + 6.61e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.75e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.13e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.58e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 752.T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.23e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.43e4T + 8.58e9T^{2} \) |
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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
−12.35212617338866797450494403577, −11.79518946580277546150482364636, −10.87974256948957506319990254921, −10.00642563938605768816981272100, −8.211691688235971490444529112248, −7.16022109745467350875703454298, −5.69299256658390621179653556730, −4.61254591917814282358678794159, −1.58789295809563625998505140196, 0,
1.58789295809563625998505140196, 4.61254591917814282358678794159, 5.69299256658390621179653556730, 7.16022109745467350875703454298, 8.211691688235971490444529112248, 10.00642563938605768816981272100, 10.87974256948957506319990254921, 11.79518946580277546150482364636, 12.35212617338866797450494403577
