| L(s) = 1 | − 0.414·3-s + 1.82·5-s − 2.82·9-s + 2.41·11-s − 2.82·13-s − 0.757·15-s + 0.171·17-s + 6.41·19-s + 5.24·23-s − 1.65·25-s + 2.41·27-s − 2.82·29-s + 5.58·31-s − 0.999·33-s + 8.65·37-s + 1.17·39-s + 6.82·41-s − 9.65·43-s − 5.17·45-s + 10.4·47-s − 0.0710·51-s − 53-s + 4.41·55-s − 2.65·57-s + 10.8·59-s − 8.65·61-s − 5.17·65-s + ⋯ |
| L(s) = 1 | − 0.239·3-s + 0.817·5-s − 0.942·9-s + 0.727·11-s − 0.784·13-s − 0.195·15-s + 0.0416·17-s + 1.47·19-s + 1.09·23-s − 0.331·25-s + 0.464·27-s − 0.525·29-s + 1.00·31-s − 0.174·33-s + 1.42·37-s + 0.187·39-s + 1.06·41-s − 1.47·43-s − 0.770·45-s + 1.51·47-s − 0.00995·51-s − 0.137·53-s + 0.595·55-s − 0.351·57-s + 1.41·59-s − 1.10·61-s − 0.641·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.775260812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.775260812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 0.171T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 8.65T + 61T^{2} \) |
| 67 | \( 1 - 2.75T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 6.07T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 1.17T + 97T^{2} \) |
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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
−9.491144051014227641421894937708, −8.823801955153168895656416136666, −7.78174163656873312642551157027, −6.96877391641899939804844047838, −6.02565676268611011052871224095, −5.46970714943462433861773733396, −4.56995359275773683854200454313, −3.25164076735164495213499023057, −2.37952981554633394661957618764, −0.981611028434021842742737928419,
0.981611028434021842742737928419, 2.37952981554633394661957618764, 3.25164076735164495213499023057, 4.56995359275773683854200454313, 5.46970714943462433861773733396, 6.02565676268611011052871224095, 6.96877391641899939804844047838, 7.78174163656873312642551157027, 8.823801955153168895656416136666, 9.491144051014227641421894937708
