| L(s) = 1 | − 729·3-s − 1.56e4·5-s + 5.52e5·7-s + 5.31e5·9-s + 1.39e6·11-s + 1.39e7·13-s + 1.13e7·15-s − 4.43e7·17-s − 7.02e7·19-s − 4.02e8·21-s − 8.40e8·23-s + 2.44e8·25-s − 3.87e8·27-s + 5.62e8·29-s + 5.49e9·31-s − 1.01e9·33-s − 8.63e9·35-s + 7.49e9·37-s − 1.01e10·39-s − 1.19e10·41-s + 7.70e9·43-s − 8.30e9·45-s − 1.04e11·47-s + 2.08e11·49-s + 3.23e10·51-s + 2.48e11·53-s − 2.18e10·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.77·7-s + 1/3·9-s + 0.237·11-s + 0.802·13-s + 0.258·15-s − 0.445·17-s − 0.342·19-s − 1.02·21-s − 1.18·23-s + 1/5·25-s − 0.192·27-s + 0.175·29-s + 1.11·31-s − 0.137·33-s − 0.793·35-s + 0.480·37-s − 0.463·39-s − 0.393·41-s + 0.185·43-s − 0.149·45-s − 1.41·47-s + 2.15·49-s + 0.257·51-s + 1.54·53-s − 0.106·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.035942407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.035942407\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{6} T \) |
| 5 | \( 1 + p^{6} T \) |
| good | 7 | \( 1 - 11276 p^{2} T + p^{13} T^{2} \) |
| 11 | \( 1 - 1396536 T + p^{13} T^{2} \) |
| 13 | \( 1 - 13961570 T + p^{13} T^{2} \) |
| 17 | \( 1 + 44320746 T + p^{13} T^{2} \) |
| 19 | \( 1 + 70293940 T + p^{13} T^{2} \) |
| 23 | \( 1 + 840907656 T + p^{13} T^{2} \) |
| 29 | \( 1 - 562865994 T + p^{13} T^{2} \) |
| 31 | \( 1 - 5491484984 T + p^{13} T^{2} \) |
| 37 | \( 1 - 7493232962 T + p^{13} T^{2} \) |
| 41 | \( 1 + 11964693990 T + p^{13} T^{2} \) |
| 43 | \( 1 - 7703197364 T + p^{13} T^{2} \) |
| 47 | \( 1 + 104453487144 T + p^{13} T^{2} \) |
| 53 | \( 1 - 248727212298 T + p^{13} T^{2} \) |
| 59 | \( 1 - 296442880584 T + p^{13} T^{2} \) |
| 61 | \( 1 - 77819243942 T + p^{13} T^{2} \) |
| 67 | \( 1 + 1177481160844 T + p^{13} T^{2} \) |
| 71 | \( 1 - 595167989280 T + p^{13} T^{2} \) |
| 73 | \( 1 - 448501295522 T + p^{13} T^{2} \) |
| 79 | \( 1 - 3092943836168 T + p^{13} T^{2} \) |
| 83 | \( 1 - 4183476685524 T + p^{13} T^{2} \) |
| 89 | \( 1 - 963708907890 T + p^{13} T^{2} \) |
| 97 | \( 1 - 10311771965858 T + p^{13} T^{2} \) |
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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
−11.95607149277648589872863654539, −11.37579523069596444329959041979, −10.39202321143647803157236702722, −8.624523249735152197180188746997, −7.78823365970603258047919866117, −6.31909638469879979522757151184, −4.95972261098975880432215193095, −4.01824028071509117718074064168, −1.97069920702789559072960703991, −0.817648145575089182495900474467,
0.817648145575089182495900474467, 1.97069920702789559072960703991, 4.01824028071509117718074064168, 4.95972261098975880432215193095, 6.31909638469879979522757151184, 7.78823365970603258047919866117, 8.624523249735152197180188746997, 10.39202321143647803157236702722, 11.37579523069596444329959041979, 11.95607149277648589872863654539
