| L(s) = 1 | + 79·5-s − 50·7-s − 121·11-s − 380·13-s + 1.15e3·17-s − 1.82e3·19-s − 3.59e3·23-s + 3.11e3·25-s − 8.03e3·29-s − 2.94e3·31-s − 3.95e3·35-s + 6.97e3·37-s + 520·41-s − 2.48e3·43-s + 6.92e3·47-s − 1.43e4·49-s + 1.37e4·53-s − 9.55e3·55-s + 3.17e4·59-s + 3.41e4·61-s − 3.00e4·65-s − 6.15e4·67-s + 1.49e4·71-s − 3.64e4·73-s + 6.05e3·77-s − 2.85e4·79-s − 7.74e4·83-s + ⋯ |
| L(s) = 1 | + 1.41·5-s − 0.385·7-s − 0.301·11-s − 0.623·13-s + 0.968·17-s − 1.15·19-s − 1.41·23-s + 0.997·25-s − 1.77·29-s − 0.550·31-s − 0.545·35-s + 0.838·37-s + 0.0483·41-s − 0.205·43-s + 0.456·47-s − 0.851·49-s + 0.670·53-s − 0.426·55-s + 1.18·59-s + 1.17·61-s − 0.881·65-s − 1.67·67-s + 0.352·71-s − 0.800·73-s + 0.116·77-s − 0.514·79-s − 1.23·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 79 T + p^{5} T^{2} \) |
| 7 | \( 1 + 50 T + p^{5} T^{2} \) |
| 13 | \( 1 + 380 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1154 T + p^{5} T^{2} \) |
| 19 | \( 1 + 96 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 3591 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8032 T + p^{5} T^{2} \) |
| 31 | \( 1 + 95 p T + p^{5} T^{2} \) |
| 37 | \( 1 - 6979 T + p^{5} T^{2} \) |
| 41 | \( 1 - 520 T + p^{5} T^{2} \) |
| 43 | \( 1 + 2486 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6920 T + p^{5} T^{2} \) |
| 53 | \( 1 - 13718 T + p^{5} T^{2} \) |
| 59 | \( 1 - 31779 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34156 T + p^{5} T^{2} \) |
| 67 | \( 1 + 61503 T + p^{5} T^{2} \) |
| 71 | \( 1 - 14971 T + p^{5} T^{2} \) |
| 73 | \( 1 + 36444 T + p^{5} T^{2} \) |
| 79 | \( 1 + 28538 T + p^{5} T^{2} \) |
| 83 | \( 1 + 77482 T + p^{5} T^{2} \) |
| 89 | \( 1 + 36271 T + p^{5} T^{2} \) |
| 97 | \( 1 + 49799 T + p^{5} T^{2} \) |
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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
−9.911588091609013567115049231733, −9.400535098957028482525819640297, −8.203905151590193298589715630096, −7.10622176849326186479276478497, −5.98071167938923516000357899507, −5.46103433558983747756893268791, −4.01469984723911304191781950560, −2.57421211616639833393004829869, −1.69371768636051967108131171102, 0,
1.69371768636051967108131171102, 2.57421211616639833393004829869, 4.01469984723911304191781950560, 5.46103433558983747756893268791, 5.98071167938923516000357899507, 7.10622176849326186479276478497, 8.203905151590193298589715630096, 9.400535098957028482525819640297, 9.911588091609013567115049231733
