| L(s) = 1 | + 3.34·3-s − 2.92·5-s + 8.21·9-s + 11-s + 5.81·13-s − 9.79·15-s + 1.21·17-s + 4.81·19-s − 9.27·23-s + 3.55·25-s + 17.4·27-s + 2.75·29-s + 1.36·31-s + 3.34·33-s − 3.89·37-s + 19.4·39-s + 10.6·41-s − 0.753·43-s − 24.0·45-s + 2.62·47-s + 4.06·51-s − 6.43·53-s − 2.92·55-s + 16.1·57-s − 6.83·59-s + 5.83·61-s − 17.0·65-s + ⋯ |
| L(s) = 1 | + 1.93·3-s − 1.30·5-s + 2.73·9-s + 0.301·11-s + 1.61·13-s − 2.52·15-s + 0.294·17-s + 1.10·19-s − 1.93·23-s + 0.711·25-s + 3.36·27-s + 0.511·29-s + 0.244·31-s + 0.582·33-s − 0.640·37-s + 3.11·39-s + 1.65·41-s − 0.114·43-s − 3.58·45-s + 0.383·47-s + 0.569·51-s − 0.884·53-s − 0.394·55-s + 2.13·57-s − 0.889·59-s + 0.747·61-s − 2.10·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.206256020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.206256020\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 13 | \( 1 - 5.81T + 13T^{2} \) |
| 17 | \( 1 - 1.21T + 17T^{2} \) |
| 19 | \( 1 - 4.81T + 19T^{2} \) |
| 23 | \( 1 + 9.27T + 23T^{2} \) |
| 29 | \( 1 - 2.75T + 29T^{2} \) |
| 31 | \( 1 - 1.36T + 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 0.753T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 + 6.43T + 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 - 5.83T + 61T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 1.09T + 79T^{2} \) |
| 83 | \( 1 - 8.68T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 1.97T + 97T^{2} \) |
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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
−7.951511295454457834553928356051, −7.44960885224703677158269134313, −6.67552012004422921334385525335, −5.79455215216678665024078369195, −4.47540893168513654550415026739, −3.97912563645104964744221325836, −3.51360407974695984820453272703, −2.91251375616414667568436011302, −1.83729444113971391709793725805, −0.969639852596352880235525834999,
0.969639852596352880235525834999, 1.83729444113971391709793725805, 2.91251375616414667568436011302, 3.51360407974695984820453272703, 3.97912563645104964744221325836, 4.47540893168513654550415026739, 5.79455215216678665024078369195, 6.67552012004422921334385525335, 7.44960885224703677158269134313, 7.951511295454457834553928356051
