| L(s) = 1 | + 2.09·3-s + 2.59·5-s − 7-s + 1.40·9-s + 0.454·11-s + 6.53·13-s + 5.45·15-s + 1.84·17-s + 5.50·19-s − 2.09·21-s + 5.88·23-s + 1.74·25-s − 3.34·27-s − 8.69·29-s − 5.69·31-s + 0.954·33-s − 2.59·35-s + 2.35·37-s + 13.7·39-s + 10.7·41-s − 0.753·43-s + 3.65·45-s − 0.465·47-s + 49-s + 3.86·51-s + 0.881·53-s + 1.18·55-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 1.16·5-s − 0.377·7-s + 0.469·9-s + 0.137·11-s + 1.81·13-s + 1.40·15-s + 0.446·17-s + 1.26·19-s − 0.458·21-s + 1.22·23-s + 0.348·25-s − 0.642·27-s − 1.61·29-s − 1.02·31-s + 0.166·33-s − 0.438·35-s + 0.387·37-s + 2.19·39-s + 1.67·41-s − 0.114·43-s + 0.545·45-s − 0.0678·47-s + 0.142·49-s + 0.541·51-s + 0.121·53-s + 0.159·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.562685899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.562685899\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2.09T + 3T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 11 | \( 1 - 0.454T + 11T^{2} \) |
| 13 | \( 1 - 6.53T + 13T^{2} \) |
| 17 | \( 1 - 1.84T + 17T^{2} \) |
| 19 | \( 1 - 5.50T + 19T^{2} \) |
| 23 | \( 1 - 5.88T + 23T^{2} \) |
| 29 | \( 1 + 8.69T + 29T^{2} \) |
| 31 | \( 1 + 5.69T + 31T^{2} \) |
| 37 | \( 1 - 2.35T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 0.753T + 43T^{2} \) |
| 47 | \( 1 + 0.465T + 47T^{2} \) |
| 53 | \( 1 - 0.881T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 8.71T + 67T^{2} \) |
| 71 | \( 1 - 0.162T + 71T^{2} \) |
| 73 | \( 1 + 3.49T + 73T^{2} \) |
| 79 | \( 1 - 8.28T + 79T^{2} \) |
| 83 | \( 1 - 3.55T + 83T^{2} \) |
| 89 | \( 1 - 1.60T + 89T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−7.903234190514897724546278826980, −7.42034481673153250434409504708, −6.48483838230221173584527544117, −5.75547049591311391273606669297, −5.38641309316436553595683937932, −4.00662621037084700344946501654, −3.42210710462079756041011962116, −2.80676059784463624112731466813, −1.84746133885732787312514480866, −1.11896822187715331983444412202,
1.11896822187715331983444412202, 1.84746133885732787312514480866, 2.80676059784463624112731466813, 3.42210710462079756041011962116, 4.00662621037084700344946501654, 5.38641309316436553595683937932, 5.75547049591311391273606669297, 6.48483838230221173584527544117, 7.42034481673153250434409504708, 7.903234190514897724546278826980
