| L(s) = 1 | − 5-s − 3.23·7-s − 2·11-s + 4.47·13-s − 4.47·17-s + 4.47·19-s + 4.76·23-s + 25-s − 2·29-s + 6.47·31-s + 3.23·35-s − 6.94·37-s − 12.4·41-s + 7.70·43-s + 7.23·47-s + 3.47·49-s + 0.472·53-s + 2·55-s − 8.47·59-s + 6·61-s − 4.47·65-s − 7.70·67-s + 2.47·71-s + 4.47·73-s + 6.47·77-s + 12.9·79-s − 3.70·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.22·7-s − 0.603·11-s + 1.24·13-s − 1.08·17-s + 1.02·19-s + 0.993·23-s + 0.200·25-s − 0.371·29-s + 1.16·31-s + 0.546·35-s − 1.14·37-s − 1.94·41-s + 1.17·43-s + 1.05·47-s + 0.496·49-s + 0.0648·53-s + 0.269·55-s − 1.10·59-s + 0.768·61-s − 0.554·65-s − 0.941·67-s + 0.293·71-s + 0.523·73-s + 0.737·77-s + 1.45·79-s − 0.407·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−7.76296485347228859662553314243, −6.88353000676477203223288685244, −6.52868974567642153735063781421, −5.62893881024335508772399412649, −4.89077693448926163411348733024, −3.88404107900890110028005616200, −3.29523348158617410245571352354, −2.55744828075660061295288842500, −1.17557750453053293088730808952, 0,
1.17557750453053293088730808952, 2.55744828075660061295288842500, 3.29523348158617410245571352354, 3.88404107900890110028005616200, 4.89077693448926163411348733024, 5.62893881024335508772399412649, 6.52868974567642153735063781421, 6.88353000676477203223288685244, 7.76296485347228859662553314243
