| L(s) = 1 | + 1.05·2-s − 3·3-s − 6.89·4-s − 17.8·5-s − 3.15·6-s + 30.1·7-s − 15.6·8-s + 9·9-s − 18.8·10-s + 50.8·11-s + 20.6·12-s + 31.7·14-s + 53.6·15-s + 38.6·16-s − 2.99·17-s + 9.46·18-s − 72.7·19-s + 123.·20-s − 90.5·21-s + 53.4·22-s − 41.9·23-s + 46.9·24-s + 195.·25-s − 27·27-s − 208.·28-s − 135.·29-s + 56.4·30-s + ⋯ |
| L(s) = 1 | + 0.371·2-s − 0.577·3-s − 0.861·4-s − 1.60·5-s − 0.214·6-s + 1.63·7-s − 0.692·8-s + 0.333·9-s − 0.594·10-s + 1.39·11-s + 0.497·12-s + 0.606·14-s + 0.923·15-s + 0.604·16-s − 0.0426·17-s + 0.123·18-s − 0.877·19-s + 1.37·20-s − 0.941·21-s + 0.518·22-s − 0.379·23-s + 0.399·24-s + 1.56·25-s − 0.192·27-s − 1.40·28-s − 0.865·29-s + 0.343·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.05T + 8T^{2} \) |
| 5 | \( 1 + 17.8T + 125T^{2} \) |
| 7 | \( 1 - 30.1T + 343T^{2} \) |
| 11 | \( 1 - 50.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 2.99T + 4.91e3T^{2} \) |
| 19 | \( 1 + 72.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 316.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 261.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 97.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 150.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 497.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 525.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 777.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 612.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 718.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 397.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 648.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 272.T + 9.12e5T^{2} \) |
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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
−10.25866254816379080457577006777, −8.826279308118305665086425585891, −8.336195061146416305617990097965, −7.43731372013740908695292398001, −6.25209003110160930979168059811, −4.88263681795202625525784446688, −4.40430840292533067822255222483, −3.62750587092704268804647267293, −1.36238775143141926564068303263, 0,
1.36238775143141926564068303263, 3.62750587092704268804647267293, 4.40430840292533067822255222483, 4.88263681795202625525784446688, 6.25209003110160930979168059811, 7.43731372013740908695292398001, 8.336195061146416305617990097965, 8.826279308118305665086425585891, 10.25866254816379080457577006777
