| L(s) = 1 | − 2.91·3-s + 3.91·5-s − 0.593·7-s + 5.51·9-s + 3.51·11-s − 2.51·13-s − 11.4·15-s + 3·17-s − 19-s + 1.73·21-s − 7.32·23-s + 10.3·25-s − 7.32·27-s − 5.73·29-s − 4.40·31-s − 10.2·33-s − 2.32·35-s + 3.42·37-s + 7.32·39-s − 10.2·41-s − 4.69·43-s + 21.5·45-s − 2.08·47-s − 6.64·49-s − 8.75·51-s − 1.08·53-s + 13.7·55-s + ⋯ |
| L(s) = 1 | − 1.68·3-s + 1.75·5-s − 0.224·7-s + 1.83·9-s + 1.05·11-s − 0.696·13-s − 2.95·15-s + 0.727·17-s − 0.229·19-s + 0.377·21-s − 1.52·23-s + 2.06·25-s − 1.40·27-s − 1.06·29-s − 0.791·31-s − 1.78·33-s − 0.392·35-s + 0.563·37-s + 1.17·39-s − 1.59·41-s − 0.716·43-s + 3.21·45-s − 0.303·47-s − 0.949·49-s − 1.22·51-s − 0.148·53-s + 1.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.91T + 3T^{2} \) |
| 5 | \( 1 - 3.91T + 5T^{2} \) |
| 7 | \( 1 + 0.593T + 7T^{2} \) |
| 11 | \( 1 - 3.51T + 11T^{2} \) |
| 13 | \( 1 + 2.51T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 + 5.73T + 29T^{2} \) |
| 31 | \( 1 + 4.40T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 4.69T + 43T^{2} \) |
| 47 | \( 1 + 2.08T + 47T^{2} \) |
| 53 | \( 1 + 1.08T + 53T^{2} \) |
| 59 | \( 1 + 8.51T + 59T^{2} \) |
| 61 | \( 1 + 7.10T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 4.40T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 0.241T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.64957914817507761044787042578, −6.81007106856841026496599058701, −6.23034854776176871552439921641, −5.86162984754760940542886302583, −5.20364495295304519049176586561, −4.50900388687325864527268546830, −3.38698322045980073024935896684, −1.95272701237168425078059313525, −1.43240787906369185966518987096, 0,
1.43240787906369185966518987096, 1.95272701237168425078059313525, 3.38698322045980073024935896684, 4.50900388687325864527268546830, 5.20364495295304519049176586561, 5.86162984754760940542886302583, 6.23034854776176871552439921641, 6.81007106856841026496599058701, 7.64957914817507761044787042578
