| L(s) = 1 | + 1.64·3-s − 1.50·5-s + 4.54·7-s − 0.283·9-s − 2.48·11-s + 2.49·13-s − 2.48·15-s + 17-s − 2.40·19-s + 7.48·21-s − 8.83·23-s − 2.72·25-s − 5.41·27-s + 7.56·29-s − 11.0·31-s − 4.09·33-s − 6.84·35-s + 1.38·37-s + 4.10·39-s − 5.32·41-s − 8.07·43-s + 0.427·45-s + 8.91·47-s + 13.6·49-s + 1.64·51-s + 7.87·53-s + 3.74·55-s + ⋯ |
| L(s) = 1 | + 0.951·3-s − 0.674·5-s + 1.71·7-s − 0.0945·9-s − 0.749·11-s + 0.691·13-s − 0.641·15-s + 0.242·17-s − 0.551·19-s + 1.63·21-s − 1.84·23-s − 0.545·25-s − 1.04·27-s + 1.40·29-s − 1.98·31-s − 0.713·33-s − 1.15·35-s + 0.227·37-s + 0.657·39-s − 0.831·41-s − 1.23·43-s + 0.0637·45-s + 1.30·47-s + 1.94·49-s + 0.230·51-s + 1.08·53-s + 0.505·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 1.64T + 3T^{2} \) |
| 5 | \( 1 + 1.50T + 5T^{2} \) |
| 7 | \( 1 - 4.54T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 - 2.49T + 13T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 + 8.83T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 - 1.38T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 47 | \( 1 - 8.91T + 47T^{2} \) |
| 53 | \( 1 - 7.87T + 53T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 5.77T + 67T^{2} \) |
| 71 | \( 1 + 5.35T + 71T^{2} \) |
| 73 | \( 1 + 5.44T + 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 9.44T + 89T^{2} \) |
| 97 | \( 1 + 3.15T + 97T^{2} \) |
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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
−7.81355999075746697168377670559, −7.14369107856575731530483924982, −5.91559800075071073209721994478, −5.42402332381248533559868306580, −4.41291646549011892239547068408, −3.98071320041599637845567358638, −3.10729661930760178008154026708, −2.15233347680622760947985240731, −1.57671948784150696527005245066, 0,
1.57671948784150696527005245066, 2.15233347680622760947985240731, 3.10729661930760178008154026708, 3.98071320041599637845567358638, 4.41291646549011892239547068408, 5.42402332381248533559868306580, 5.91559800075071073209721994478, 7.14369107856575731530483924982, 7.81355999075746697168377670559
