L(s) = 1 | + 25·5-s − 62·7-s + 144·11-s − 654·13-s + 1.19e3·17-s + 556·19-s − 2.18e3·23-s + 625·25-s + 1.57e3·29-s + 9.66e3·31-s − 1.55e3·35-s − 3.53e3·37-s − 7.46e3·41-s − 7.11e3·43-s + 2.82e4·47-s − 1.29e4·49-s + 1.30e4·53-s + 3.60e3·55-s + 3.70e4·59-s + 3.95e4·61-s − 1.63e4·65-s − 5.67e4·67-s − 4.55e4·71-s + 1.18e4·73-s − 8.92e3·77-s + 9.42e4·79-s + 3.14e4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.478·7-s + 0.358·11-s − 1.07·13-s + 0.998·17-s + 0.353·19-s − 0.860·23-s + 1/5·25-s + 0.348·29-s + 1.80·31-s − 0.213·35-s − 0.424·37-s − 0.693·41-s − 0.586·43-s + 1.86·47-s − 0.771·49-s + 0.637·53-s + 0.160·55-s + 1.38·59-s + 1.36·61-s − 0.479·65-s − 1.54·67-s − 1.07·71-s + 0.260·73-s − 0.171·77-s + 1.69·79-s + 0.501·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.074348642\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.074348642\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
good | 7 | \( 1 + 62 T + p^{5} T^{2} \) |
| 11 | \( 1 - 144 T + p^{5} T^{2} \) |
| 13 | \( 1 + 654 T + p^{5} T^{2} \) |
| 17 | \( 1 - 70 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 556 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2182 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1578 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9660 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3534 T + p^{5} T^{2} \) |
| 41 | \( 1 + 182 p T + p^{5} T^{2} \) |
| 43 | \( 1 + 7114 T + p^{5} T^{2} \) |
| 47 | \( 1 - 602 p T + p^{5} T^{2} \) |
| 53 | \( 1 - 13046 T + p^{5} T^{2} \) |
| 59 | \( 1 - 37092 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39570 T + p^{5} T^{2} \) |
| 67 | \( 1 + 56734 T + p^{5} T^{2} \) |
| 71 | \( 1 + 45588 T + p^{5} T^{2} \) |
| 73 | \( 1 - 11842 T + p^{5} T^{2} \) |
| 79 | \( 1 - 94216 T + p^{5} T^{2} \) |
| 83 | \( 1 - 31482 T + p^{5} T^{2} \) |
| 89 | \( 1 - 94054 T + p^{5} T^{2} \) |
| 97 | \( 1 - 23714 T + p^{5} T^{2} \) |
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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
−10.19131959147977562580842439262, −9.971857605728331527751048099259, −8.848553775367841748859549177250, −7.75690650264061333366615703474, −6.76385430739572280141648619794, −5.79989847540953640044860870936, −4.74460007654650185417604275880, −3.40639657220047482603143127182, −2.24856837449652082567304116334, −0.78044757537213289225227121995,
0.78044757537213289225227121995, 2.24856837449652082567304116334, 3.40639657220047482603143127182, 4.74460007654650185417604275880, 5.79989847540953640044860870936, 6.76385430739572280141648619794, 7.75690650264061333366615703474, 8.848553775367841748859549177250, 9.971857605728331527751048099259, 10.19131959147977562580842439262