| L(s) = 1 | + 5-s + 4.57·11-s − 1.74·13-s + 2.47·17-s − 7.73·19-s + 5.99·23-s + 25-s + 3.49·29-s − 7.07·31-s − 10.9·37-s − 12.4·41-s − 10.4·43-s − 8.47·47-s − 11.6·53-s + 4.57·55-s − 4.94·59-s + 4.91·61-s − 1.74·65-s + 4.94·67-s − 7.40·71-s + 3.90·73-s + 10.9·79-s + 9.41·83-s + 2.47·85-s − 6·89-s − 7.73·95-s + 13.7·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.37·11-s − 0.484·13-s + 0.599·17-s − 1.77·19-s + 1.24·23-s + 0.200·25-s + 0.649·29-s − 1.27·31-s − 1.79·37-s − 1.94·41-s − 1.59·43-s − 1.23·47-s − 1.59·53-s + 0.617·55-s − 0.643·59-s + 0.628·61-s − 0.216·65-s + 0.604·67-s − 0.878·71-s + 0.457·73-s + 1.23·79-s + 1.03·83-s + 0.268·85-s − 0.635·89-s − 0.793·95-s + 1.39·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 7.73T + 19T^{2} \) |
| 23 | \( 1 - 5.99T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 8.47T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 4.91T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 - 3.90T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 9.41T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.20720779010033908122376088700, −6.62048734168842986372638772743, −6.31199429782569229822937897052, −5.12204563754807290726132523536, −4.84245327660361332400291039337, −3.70042507837242583012471022759, −3.22817273620760739238122135566, −1.97775226833603758850958942030, −1.45779738903558511372901745290, 0,
1.45779738903558511372901745290, 1.97775226833603758850958942030, 3.22817273620760739238122135566, 3.70042507837242583012471022759, 4.84245327660361332400291039337, 5.12204563754807290726132523536, 6.31199429782569229822937897052, 6.62048734168842986372638772743, 7.20720779010033908122376088700
