| L(s) = 1 | + 0.399·5-s + 2.09·7-s − 4.31·11-s + 4.27·13-s + 5.86·17-s − 0.918·19-s + 7.26·23-s − 4.84·25-s + 8.94·29-s − 3.68·31-s + 0.838·35-s − 8.80·37-s − 1.00·41-s + 7.32·43-s + 3.15·47-s − 2.60·49-s + 3.31·53-s − 1.72·55-s + 0.721·59-s − 12.2·61-s + 1.71·65-s + 11.0·67-s + 13.2·71-s − 12.1·73-s − 9.04·77-s + 0.0176·79-s + 2.24·83-s + ⋯ |
| L(s) = 1 | + 0.178·5-s + 0.792·7-s − 1.30·11-s + 1.18·13-s + 1.42·17-s − 0.210·19-s + 1.51·23-s − 0.968·25-s + 1.66·29-s − 0.661·31-s + 0.141·35-s − 1.44·37-s − 0.156·41-s + 1.11·43-s + 0.460·47-s − 0.371·49-s + 0.455·53-s − 0.232·55-s + 0.0939·59-s − 1.56·61-s + 0.212·65-s + 1.35·67-s + 1.57·71-s − 1.42·73-s − 1.03·77-s + 0.00198·79-s + 0.246·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.427273204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427273204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 0.399T + 5T^{2} \) |
| 7 | \( 1 - 2.09T + 7T^{2} \) |
| 11 | \( 1 + 4.31T + 11T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 17 | \( 1 - 5.86T + 17T^{2} \) |
| 19 | \( 1 + 0.918T + 19T^{2} \) |
| 23 | \( 1 - 7.26T + 23T^{2} \) |
| 29 | \( 1 - 8.94T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 + 8.80T + 37T^{2} \) |
| 41 | \( 1 + 1.00T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 - 3.31T + 53T^{2} \) |
| 59 | \( 1 - 0.721T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 0.0176T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 5.28T + 89T^{2} \) |
| 97 | \( 1 - 7.03T + 97T^{2} \) |
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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
−8.108740736811150031158980425788, −7.50767161556319758962958012872, −6.70448121491870876289017005225, −5.72346877809155372687378577948, −5.32741314522107742421754442465, −4.56582722624540627524254682998, −3.54303858986378400066442207040, −2.83898802594661297102179435368, −1.77453456910533170980037529563, −0.859522477875055375333556130384,
0.859522477875055375333556130384, 1.77453456910533170980037529563, 2.83898802594661297102179435368, 3.54303858986378400066442207040, 4.56582722624540627524254682998, 5.32741314522107742421754442465, 5.72346877809155372687378577948, 6.70448121491870876289017005225, 7.50767161556319758962958012872, 8.108740736811150031158980425788
