| L(s) = 1 | − 7.02·3-s − 2.91·5-s + 14.5·7-s + 22.2·9-s + 11.3·11-s + 48.9·13-s + 20.4·15-s − 71.5·19-s − 101.·21-s − 37.9·23-s − 116.·25-s + 33.0·27-s − 76.9·29-s + 65.2·31-s − 79.5·33-s − 42.2·35-s − 78.6·37-s − 343.·39-s + 358.·41-s − 82.9·43-s − 65.0·45-s − 11.8·47-s − 132.·49-s − 565.·53-s − 33.0·55-s + 502.·57-s − 335.·59-s + ⋯ |
| L(s) = 1 | − 1.35·3-s − 0.260·5-s + 0.783·7-s + 0.825·9-s + 0.310·11-s + 1.04·13-s + 0.352·15-s − 0.863·19-s − 1.05·21-s − 0.343·23-s − 0.932·25-s + 0.235·27-s − 0.492·29-s + 0.377·31-s − 0.419·33-s − 0.204·35-s − 0.349·37-s − 1.41·39-s + 1.36·41-s − 0.294·43-s − 0.215·45-s − 0.0366·47-s − 0.386·49-s − 1.46·53-s − 0.0810·55-s + 1.16·57-s − 0.739·59-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 | \( 1 + 7.02T + 27T^{2} \) |
| 5 | \( 1 + 2.91T + 125T^{2} \) |
| 7 | \( 1 - 14.5T + 343T^{2} \) |
| 11 | \( 1 - 11.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.9T + 2.19e3T^{2} \) |
| 19 | \( 1 + 71.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 37.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 65.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 358.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 82.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 11.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 565.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 335.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 344.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 634.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 58.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 229.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.42e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 144.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 426.T + 9.12e5T^{2} \) |
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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
−8.199261119125628315445913748641, −7.48865445922610084148018882594, −6.35260863622054478232247032325, −6.08154099888953140684289786624, −5.11019968362497463363650513969, −4.40137471175787599260237851142, −3.57386757850543763550468749186, −2.02708686289751742405888693785, −1.05747711168196537961175084267, 0,
1.05747711168196537961175084267, 2.02708686289751742405888693785, 3.57386757850543763550468749186, 4.40137471175787599260237851142, 5.11019968362497463363650513969, 6.08154099888953140684289786624, 6.35260863622054478232247032325, 7.48865445922610084148018882594, 8.199261119125628315445913748641
