| L(s) = 1 | + 5-s + 0.763·11-s − 13-s − 2·17-s − 0.763·19-s − 3.23·23-s + 25-s − 8.47·29-s + 5.70·31-s − 8.47·37-s + 10.9·41-s + 3.23·43-s + 12.9·47-s − 7·49-s + 10.9·53-s + 0.763·55-s − 5.70·59-s − 4.47·61-s − 65-s − 10.4·67-s − 0.763·71-s − 7.52·73-s − 6.47·79-s + 4·83-s − 2·85-s − 10·89-s − 0.763·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.230·11-s − 0.277·13-s − 0.485·17-s − 0.175·19-s − 0.674·23-s + 0.200·25-s − 1.57·29-s + 1.02·31-s − 1.39·37-s + 1.70·41-s + 0.493·43-s + 1.88·47-s − 49-s + 1.50·53-s + 0.103·55-s − 0.743·59-s − 0.572·61-s − 0.124·65-s − 1.27·67-s − 0.0906·71-s − 0.881·73-s − 0.728·79-s + 0.439·83-s − 0.216·85-s − 1.05·89-s − 0.0783·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 0.763T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 5.70T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 + 6.47T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.32528465495988304631423541998, −6.72522284600748965856639734297, −5.85826773704523250404020558786, −5.52689218429173939399842681449, −4.43216327111233501983180769240, −4.00267098349224159635694869177, −2.90714493868132380909683568780, −2.20238782359475020367649432397, −1.31640764684290981613069667801, 0,
1.31640764684290981613069667801, 2.20238782359475020367649432397, 2.90714493868132380909683568780, 4.00267098349224159635694869177, 4.43216327111233501983180769240, 5.52689218429173939399842681449, 5.85826773704523250404020558786, 6.72522284600748965856639734297, 7.32528465495988304631423541998
