| L(s) = 1 | − 9·3-s + 48.1·5-s − 213.·7-s + 81·9-s + 618.·11-s + 37.0·13-s − 433.·15-s + 1.77e3·17-s + 2.14e3·19-s + 1.92e3·21-s − 1.66e3·23-s − 809·25-s − 729·27-s − 2.80e3·29-s − 9.32e3·31-s − 5.56e3·33-s − 1.02e4·35-s + 9.42e3·37-s − 333.·39-s − 2.45e3·41-s + 6.29e3·43-s + 3.89e3·45-s − 1.84e4·47-s + 2.88e4·49-s − 1.59e4·51-s + 2.44e4·53-s + 2.97e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.860·5-s − 1.64·7-s + 0.333·9-s + 1.54·11-s + 0.0608·13-s − 0.497·15-s + 1.49·17-s + 1.36·19-s + 0.951·21-s − 0.657·23-s − 0.258·25-s − 0.192·27-s − 0.618·29-s − 1.74·31-s − 0.890·33-s − 1.41·35-s + 1.13·37-s − 0.0351·39-s − 0.228·41-s + 0.519·43-s + 0.286·45-s − 1.21·47-s + 1.71·49-s − 0.860·51-s + 1.19·53-s + 1.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.934754726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.934754726\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| good | 5 | \( 1 - 48.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 213.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 618.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 37.0T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.77e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.14e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.66e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.42e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.29e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.84e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.44e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.72e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.82e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.03e4T + 8.58e9T^{2} \) |
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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
−9.613350708191551127338397204971, −9.203088064580821925099204298970, −7.60792852394492845397848915784, −6.75850079213785476780858879949, −5.95792767310435404666733401280, −5.52823116477786895300228293423, −3.90054814492866401727182458567, −3.22378932907945260129151779163, −1.70440543373699357828278191429, −0.68228519121032172423086666040,
0.68228519121032172423086666040, 1.70440543373699357828278191429, 3.22378932907945260129151779163, 3.90054814492866401727182458567, 5.52823116477786895300228293423, 5.95792767310435404666733401280, 6.75850079213785476780858879949, 7.60792852394492845397848915784, 9.203088064580821925099204298970, 9.613350708191551127338397204971
