| L(s) = 1 | + 243·3-s + 9.99e3·5-s + 8.61e4·7-s + 5.90e4·9-s + 8.06e5·11-s − 9.60e5·13-s + 2.42e6·15-s − 4.30e6·17-s − 4.01e5·19-s + 2.09e7·21-s − 1.77e7·23-s + 5.09e7·25-s + 1.43e7·27-s − 8.47e7·29-s − 1.40e8·31-s + 1.95e8·33-s + 8.60e8·35-s − 4.13e8·37-s − 2.33e8·39-s + 1.50e8·41-s − 7.06e8·43-s + 5.89e8·45-s + 2.47e9·47-s + 5.44e9·49-s − 1.04e9·51-s + 1.60e9·53-s + 8.05e9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.42·5-s + 1.93·7-s + 1/3·9-s + 1.50·11-s − 0.717·13-s + 0.825·15-s − 0.735·17-s − 0.0371·19-s + 1.11·21-s − 0.575·23-s + 1.04·25-s + 0.192·27-s − 0.766·29-s − 0.884·31-s + 0.871·33-s + 2.76·35-s − 0.980·37-s − 0.414·39-s + 0.202·41-s − 0.733·43-s + 0.476·45-s + 1.57·47-s + 2.75·49-s − 0.424·51-s + 0.525·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.185825414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.185825414\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 - 1998 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 12304 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 806004 T + p^{11} T^{2} \) |
| 13 | \( 1 + 960250 T + p^{11} T^{2} \) |
| 17 | \( 1 + 4306878 T + p^{11} T^{2} \) |
| 19 | \( 1 + 401300 T + p^{11} T^{2} \) |
| 23 | \( 1 + 17751528 T + p^{11} T^{2} \) |
| 29 | \( 1 + 84704994 T + p^{11} T^{2} \) |
| 31 | \( 1 + 140930504 T + p^{11} T^{2} \) |
| 37 | \( 1 + 413506594 T + p^{11} T^{2} \) |
| 41 | \( 1 - 150094890 T + p^{11} T^{2} \) |
| 43 | \( 1 + 706702028 T + p^{11} T^{2} \) |
| 47 | \( 1 - 2475725472 T + p^{11} T^{2} \) |
| 53 | \( 1 - 30191022 p T + p^{11} T^{2} \) |
| 59 | \( 1 + 3945492396 T + p^{11} T^{2} \) |
| 61 | \( 1 + 885973498 T + p^{11} T^{2} \) |
| 67 | \( 1 - 4881597772 T + p^{11} T^{2} \) |
| 71 | \( 1 + 12631469400 T + p^{11} T^{2} \) |
| 73 | \( 1 - 1423335194 T + p^{11} T^{2} \) |
| 79 | \( 1 + 667407512 T + p^{11} T^{2} \) |
| 83 | \( 1 + 5716071828 T + p^{11} T^{2} \) |
| 89 | \( 1 + 85738736790 T + p^{11} T^{2} \) |
| 97 | \( 1 + 52302647806 T + p^{11} T^{2} \) |
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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
−13.67342401155284965635680420893, −12.07750045323422711968605969953, −10.85171253129549972427151575017, −9.489465067700148324627991200333, −8.601732069564129899791627023741, −7.12784884010910326039248951236, −5.56288352631606914860546587457, −4.27221440872707234436884652512, −2.11123532368037855273053601268, −1.49917451241540478608357234752,
1.49917451241540478608357234752, 2.11123532368037855273053601268, 4.27221440872707234436884652512, 5.56288352631606914860546587457, 7.12784884010910326039248951236, 8.601732069564129899791627023741, 9.489465067700148324627991200333, 10.85171253129549972427151575017, 12.07750045323422711968605969953, 13.67342401155284965635680420893
