| L(s) = 1 | + 0.409·3-s + 15.5·5-s + 8.71·7-s − 26.8·9-s − 45.9·11-s + 33.8·13-s + 6.38·15-s − 63.7·19-s + 3.56·21-s − 39.2·23-s + 118.·25-s − 22.0·27-s − 116.·29-s + 175.·31-s − 18.8·33-s + 135.·35-s + 442.·37-s + 13.8·39-s + 276.·41-s − 432.·43-s − 418.·45-s + 195.·47-s − 267.·49-s − 196.·53-s − 716.·55-s − 26.0·57-s − 410.·59-s + ⋯ |
| L(s) = 1 | + 0.0787·3-s + 1.39·5-s + 0.470·7-s − 0.993·9-s − 1.25·11-s + 0.722·13-s + 0.109·15-s − 0.769·19-s + 0.0370·21-s − 0.355·23-s + 0.946·25-s − 0.157·27-s − 0.745·29-s + 1.01·31-s − 0.0991·33-s + 0.656·35-s + 1.96·37-s + 0.0568·39-s + 1.05·41-s − 1.53·43-s − 1.38·45-s + 0.608·47-s − 0.778·49-s − 0.508·53-s − 1.75·55-s − 0.0606·57-s − 0.906·59-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 | \( 1 - 0.409T + 27T^{2} \) |
| 5 | \( 1 - 15.5T + 125T^{2} \) |
| 7 | \( 1 - 8.71T + 343T^{2} \) |
| 11 | \( 1 + 45.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.8T + 2.19e3T^{2} \) |
| 19 | \( 1 + 63.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 39.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 116.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 442.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 432.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 195.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 196.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 410.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 508.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 422.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 800.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 473.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 353.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 300.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 296.T + 9.12e5T^{2} \) |
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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
−8.215934916932923523025384768945, −7.76115257113232467217202226156, −6.35566860109865707615871813606, −5.96772909449143063117110048240, −5.25820056245022170729634273307, −4.38004919560822267762302193998, −2.98035988542285003178584957482, −2.36673048244954624494839053501, −1.41824999551630602169861223930, 0,
1.41824999551630602169861223930, 2.36673048244954624494839053501, 2.98035988542285003178584957482, 4.38004919560822267762302193998, 5.25820056245022170729634273307, 5.96772909449143063117110048240, 6.35566860109865707615871813606, 7.76115257113232467217202226156, 8.215934916932923523025384768945
