| L(s) = 1 | + 1.53·3-s − 1.87·5-s + 2.06·7-s − 0.652·9-s − 2.06·11-s − 1.53·13-s − 2.87·15-s + 0.347·17-s + 3.16·21-s + 2.94·23-s − 1.46·25-s − 5.59·27-s − 10.4·29-s + 5.45·31-s − 3.16·33-s − 3.87·35-s − 5.51·37-s − 2.34·39-s + 9.98·41-s − 6.94·43-s + 1.22·45-s + 5.17·47-s − 2.73·49-s + 0.532·51-s − 7.76·53-s + 3.87·55-s + 0.263·59-s + ⋯ |
| L(s) = 1 | + 0.884·3-s − 0.840·5-s + 0.780·7-s − 0.217·9-s − 0.622·11-s − 0.424·13-s − 0.743·15-s + 0.0842·17-s + 0.690·21-s + 0.613·23-s − 0.293·25-s − 1.07·27-s − 1.94·29-s + 0.979·31-s − 0.550·33-s − 0.655·35-s − 0.907·37-s − 0.375·39-s + 1.55·41-s − 1.05·43-s + 0.182·45-s + 0.754·47-s − 0.391·49-s + 0.0745·51-s − 1.06·53-s + 0.523·55-s + 0.0343·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 5 | \( 1 + 1.87T + 5T^{2} \) |
| 7 | \( 1 - 2.06T + 7T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 0.347T + 17T^{2} \) |
| 23 | \( 1 - 2.94T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 5.45T + 31T^{2} \) |
| 37 | \( 1 + 5.51T + 37T^{2} \) |
| 41 | \( 1 - 9.98T + 41T^{2} \) |
| 43 | \( 1 + 6.94T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 + 7.76T + 53T^{2} \) |
| 59 | \( 1 - 0.263T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 7.04T + 67T^{2} \) |
| 71 | \( 1 - 6.46T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 4.59T + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−8.311948651069349744819767551472, −7.64081839791476466401844205292, −7.39168005416505466723638625154, −6.03918749804845099822220476770, −5.16875187628295276348036339993, −4.37130543938252912023377598279, −3.48266684265518241467012949813, −2.70689353041491739251184309421, −1.70503818560569579679607702043, 0,
1.70503818560569579679607702043, 2.70689353041491739251184309421, 3.48266684265518241467012949813, 4.37130543938252912023377598279, 5.16875187628295276348036339993, 6.03918749804845099822220476770, 7.39168005416505466723638625154, 7.64081839791476466401844205292, 8.311948651069349744819767551472
