| L(s) = 1 | + 9·3-s + 33.4·5-s + 81·9-s − 707.·11-s + 83.0·13-s + 300.·15-s + 512.·17-s + 1.03e3·19-s + 565.·23-s − 2.00e3·25-s + 729·27-s + 81.1·29-s − 2.27e3·31-s − 6.36e3·33-s − 1.01e4·37-s + 747.·39-s + 4.26e3·41-s − 1.83e4·43-s + 2.70e3·45-s − 2.25e4·47-s + 4.61e3·51-s − 3.59e3·53-s − 2.36e4·55-s + 9.31e3·57-s + 2.00e4·59-s + 3.29e4·61-s + 2.77e3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.597·5-s + 0.333·9-s − 1.76·11-s + 0.136·13-s + 0.345·15-s + 0.429·17-s + 0.657·19-s + 0.222·23-s − 0.642·25-s + 0.192·27-s + 0.0179·29-s − 0.424·31-s − 1.01·33-s − 1.21·37-s + 0.0786·39-s + 0.395·41-s − 1.51·43-s + 0.199·45-s − 1.48·47-s + 0.248·51-s − 0.175·53-s − 1.05·55-s + 0.379·57-s + 0.748·59-s + 1.13·61-s + 0.0814·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 33.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 707.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 83.0T + 3.71e5T^{2} \) |
| 17 | \( 1 - 512.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 565.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 81.1T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.27e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.83e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.25e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.59e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 130.T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.14e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.62e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.04e4T + 8.58e9T^{2} \) |
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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
−9.646816452145910773736916351992, −8.490071103635805693915145297706, −7.83177888431172913346278634242, −6.90608167240432144034535807618, −5.63165112379477291361083766472, −4.99444763709064264923097127847, −3.51925886232241659877536443295, −2.60909979313089803152006306619, −1.57162147693387045899322367371, 0,
1.57162147693387045899322367371, 2.60909979313089803152006306619, 3.51925886232241659877536443295, 4.99444763709064264923097127847, 5.63165112379477291361083766472, 6.90608167240432144034535807618, 7.83177888431172913346278634242, 8.490071103635805693915145297706, 9.646816452145910773736916351992
