| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s + 4·5-s − 4·6-s + 9-s + 8·10-s − 3·11-s − 4·12-s − 8·15-s − 4·16-s − 3·17-s + 2·18-s + 2·19-s + 8·20-s − 6·22-s − 23-s + 11·25-s + 4·27-s − 6·29-s − 16·30-s + 31-s − 8·32-s + 6·33-s − 6·34-s + 2·36-s + 4·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s + 1.78·5-s − 1.63·6-s + 1/3·9-s + 2.52·10-s − 0.904·11-s − 1.15·12-s − 2.06·15-s − 16-s − 0.727·17-s + 0.471·18-s + 0.458·19-s + 1.78·20-s − 1.27·22-s − 0.208·23-s + 11/5·25-s + 0.769·27-s − 1.11·29-s − 2.92·30-s + 0.179·31-s − 1.41·32-s + 1.04·33-s − 1.02·34-s + 1/3·36-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7267 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−6.92118217792216295945391535240, −6.57651216510585968023959662320, −5.83511537723293853376460210660, −5.40893162679170813082942522302, −5.10232176291010875850962281072, −4.28214561130586735313193239905, −3.10890097960520390641851040111, −2.44182331693831212333222760548, −1.56891755503657286870113571467, 0,
1.56891755503657286870113571467, 2.44182331693831212333222760548, 3.10890097960520390641851040111, 4.28214561130586735313193239905, 5.10232176291010875850962281072, 5.40893162679170813082942522302, 5.83511537723293853376460210660, 6.57651216510585968023959662320, 6.92118217792216295945391535240
