| L(s) = 1 | − 1.79·2-s − 0.144·3-s + 1.23·4-s − 4.04·5-s + 0.259·6-s + 1.37·8-s − 2.97·9-s + 7.26·10-s − 4.40·11-s − 0.177·12-s − 5.70·13-s + 0.582·15-s − 4.94·16-s − 6.08·17-s + 5.35·18-s + 4.26·19-s − 4.98·20-s + 7.91·22-s + 5.72·23-s − 0.198·24-s + 11.3·25-s + 10.2·26-s + 0.861·27-s + 4.10·29-s − 1.04·30-s − 8.35·31-s + 6.13·32-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.0832·3-s + 0.616·4-s − 1.80·5-s + 0.105·6-s + 0.487·8-s − 0.993·9-s + 2.29·10-s − 1.32·11-s − 0.0512·12-s − 1.58·13-s + 0.150·15-s − 1.23·16-s − 1.47·17-s + 1.26·18-s + 0.979·19-s − 1.11·20-s + 1.68·22-s + 1.19·23-s − 0.0405·24-s + 2.26·25-s + 2.01·26-s + 0.165·27-s + 0.761·29-s − 0.191·30-s − 1.50·31-s + 1.08·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 3 | \( 1 + 0.144T + 3T^{2} \) |
| 5 | \( 1 + 4.04T + 5T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 + 6.08T + 17T^{2} \) |
| 19 | \( 1 - 4.26T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 - 4.10T + 29T^{2} \) |
| 31 | \( 1 + 8.35T + 31T^{2} \) |
| 37 | \( 1 - 0.325T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 - 0.975T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.335T + 53T^{2} \) |
| 59 | \( 1 - 6.71T + 59T^{2} \) |
| 61 | \( 1 - 2.94T + 61T^{2} \) |
| 67 | \( 1 - 6.58T + 67T^{2} \) |
| 71 | \( 1 + 4.46T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 1.54T + 79T^{2} \) |
| 83 | \( 1 - 2.17T + 83T^{2} \) |
| 89 | \( 1 + 7.47T + 89T^{2} \) |
| 97 | \( 1 + 7.61T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84872739919285972632374393588, −7.17055329595092364928693364706, −6.93865697959356383527624779459, −5.27138441270865178165288099839, −4.93904300629388232853904869774, −4.04012647986725229827632083015, −2.94740532571757952165723315110, −2.35644303250282586883634709625, −0.63839446698085633150497627870, 0,
0.63839446698085633150497627870, 2.35644303250282586883634709625, 2.94740532571757952165723315110, 4.04012647986725229827632083015, 4.93904300629388232853904869774, 5.27138441270865178165288099839, 6.93865697959356383527624779459, 7.17055329595092364928693364706, 7.84872739919285972632374393588
