| L(s) = 1 | − 6.63·3-s + 3.97·5-s − 24.2·7-s + 17.0·9-s − 14.5·11-s + 62.7·13-s − 26.3·15-s + 12.7·19-s + 160.·21-s − 67.4·23-s − 109.·25-s + 65.9·27-s − 146.·29-s − 211.·31-s + 96.6·33-s − 96.1·35-s + 313.·37-s − 416.·39-s + 12.3·41-s + 300.·43-s + 67.7·45-s + 65.4·47-s + 242.·49-s + 610.·53-s − 57.7·55-s − 84.9·57-s − 152.·59-s + ⋯ |
| L(s) = 1 | − 1.27·3-s + 0.355·5-s − 1.30·7-s + 0.631·9-s − 0.398·11-s + 1.33·13-s − 0.453·15-s + 0.154·19-s + 1.66·21-s − 0.611·23-s − 0.873·25-s + 0.470·27-s − 0.936·29-s − 1.22·31-s + 0.509·33-s − 0.464·35-s + 1.39·37-s − 1.71·39-s + 0.0472·41-s + 1.06·43-s + 0.224·45-s + 0.203·47-s + 0.708·49-s + 1.58·53-s − 0.141·55-s − 0.197·57-s − 0.337·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 6.63T + 27T^{2} \) |
| 5 | \( 1 - 3.97T + 125T^{2} \) |
| 7 | \( 1 + 24.2T + 343T^{2} \) |
| 11 | \( 1 + 14.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 12.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 146.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 211.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 12.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 300.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 65.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 610.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 152.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 174.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 525.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 837.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 494.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 885.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 850.T + 9.12e5T^{2} \) |
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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
−8.261582301591786896953845177672, −7.25478826458362631749815930965, −6.47209494288156807497007821105, −5.74288850312272760853340686765, −5.62288666345433913198819618652, −4.19178130144609242425488903855, −3.46352666394957038927523094161, −2.24791150073735221474811677757, −0.909849482220411660533969754422, 0,
0.909849482220411660533969754422, 2.24791150073735221474811677757, 3.46352666394957038927523094161, 4.19178130144609242425488903855, 5.62288666345433913198819618652, 5.74288850312272760853340686765, 6.47209494288156807497007821105, 7.25478826458362631749815930965, 8.261582301591786896953845177672
