| L(s) = 1 | − 2.13·3-s + 0.368·5-s − 4.29·7-s + 1.55·9-s − 11-s + 4.29·13-s − 0.785·15-s + 1.76·17-s − 19-s + 9.16·21-s + 2.34·23-s − 4.86·25-s + 3.09·27-s − 1.93·29-s + 6.63·31-s + 2.13·33-s − 1.58·35-s − 2.57·37-s − 9.16·39-s + 3.79·41-s + 11.7·43-s + 0.571·45-s − 8.32·47-s + 11.4·49-s − 3.76·51-s + 7.31·53-s − 0.368·55-s + ⋯ |
| L(s) = 1 | − 1.23·3-s + 0.164·5-s − 1.62·7-s + 0.516·9-s − 0.301·11-s + 1.19·13-s − 0.202·15-s + 0.428·17-s − 0.229·19-s + 1.99·21-s + 0.488·23-s − 0.972·25-s + 0.594·27-s − 0.359·29-s + 1.19·31-s + 0.371·33-s − 0.267·35-s − 0.424·37-s − 1.46·39-s + 0.592·41-s + 1.78·43-s + 0.0851·45-s − 1.21·47-s + 1.63·49-s − 0.527·51-s + 1.00·53-s − 0.0496·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.13T + 3T^{2} \) |
| 5 | \( 1 - 0.368T + 5T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 - 6.63T + 31T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 - 7.31T + 53T^{2} \) |
| 59 | \( 1 - 0.635T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 5.63T + 67T^{2} \) |
| 71 | \( 1 - 2.65T + 71T^{2} \) |
| 73 | \( 1 - 0.519T + 73T^{2} \) |
| 79 | \( 1 - 3.02T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 4.05T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.272871650683284500904704075741, −7.26324808973496570761491227623, −6.47826516649177932583030674492, −5.99844453539041185586733270766, −5.55225609584244122519726172287, −4.41398998672318018402272843604, −3.52845650081143516568760419031, −2.67284895567722011160111067536, −1.10117988099581115066112756228, 0,
1.10117988099581115066112756228, 2.67284895567722011160111067536, 3.52845650081143516568760419031, 4.41398998672318018402272843604, 5.55225609584244122519726172287, 5.99844453539041185586733270766, 6.47826516649177932583030674492, 7.26324808973496570761491227623, 8.272871650683284500904704075741
