| L(s) = 1 | − 3.19·3-s + 3.86·5-s + 0.0940·7-s + 7.22·9-s + 0.580·11-s + 6.55·13-s − 12.3·15-s − 0.0338·17-s + 19-s − 0.300·21-s − 0.0763·23-s + 9.91·25-s − 13.5·27-s + 9.97·29-s + 5.37·31-s − 1.85·33-s + 0.363·35-s + 4.48·37-s − 20.9·39-s − 6.68·41-s + 5.10·43-s + 27.8·45-s + 2.83·47-s − 6.99·49-s + 0.108·51-s + 12.2·53-s + 2.24·55-s + ⋯ |
| L(s) = 1 | − 1.84·3-s + 1.72·5-s + 0.0355·7-s + 2.40·9-s + 0.174·11-s + 1.81·13-s − 3.18·15-s − 0.00821·17-s + 0.229·19-s − 0.0656·21-s − 0.0159·23-s + 1.98·25-s − 2.59·27-s + 1.85·29-s + 0.965·31-s − 0.323·33-s + 0.0614·35-s + 0.737·37-s − 3.35·39-s − 1.04·41-s + 0.778·43-s + 4.15·45-s + 0.413·47-s − 0.998·49-s + 0.0151·51-s + 1.68·53-s + 0.302·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.992825851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992825851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 3.19T + 3T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 7 | \( 1 - 0.0940T + 7T^{2} \) |
| 11 | \( 1 - 0.580T + 11T^{2} \) |
| 13 | \( 1 - 6.55T + 13T^{2} \) |
| 17 | \( 1 + 0.0338T + 17T^{2} \) |
| 23 | \( 1 + 0.0763T + 23T^{2} \) |
| 29 | \( 1 - 9.97T + 29T^{2} \) |
| 31 | \( 1 - 5.37T + 31T^{2} \) |
| 37 | \( 1 - 4.48T + 37T^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 - 5.10T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 83 | \( 1 + 5.74T + 83T^{2} \) |
| 89 | \( 1 - 3.86T + 89T^{2} \) |
| 97 | \( 1 - 8.62T + 97T^{2} \) |
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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
−8.090203662587756736131153380113, −6.81626338261524415090075869230, −6.42099832791348142386792339802, −6.04027655819519936311667097445, −5.38454517626523177010766619699, −4.79237333517997515225115161493, −3.88934030223973921254396336547, −2.58714453982951605818353274898, −1.37939644089496350085932627432, −0.980068070993594761023926900369,
0.980068070993594761023926900369, 1.37939644089496350085932627432, 2.58714453982951605818353274898, 3.88934030223973921254396336547, 4.79237333517997515225115161493, 5.38454517626523177010766619699, 6.04027655819519936311667097445, 6.42099832791348142386792339802, 6.81626338261524415090075869230, 8.090203662587756736131153380113
