| L(s) = 1 | + 29.2·3-s + 98.5·5-s + 610.·9-s − 129.·11-s + 126.·13-s + 2.87e3·15-s − 1.61e3·17-s + 1.39e3·19-s − 2.77e3·23-s + 6.57e3·25-s + 1.07e4·27-s + 3.49e3·29-s + 710.·31-s − 3.77e3·33-s − 3.05e3·37-s + 3.68e3·39-s + 1.52e4·41-s − 7.88e3·43-s + 6.01e4·45-s + 1.67e4·47-s − 4.71e4·51-s − 3.23e3·53-s − 1.27e4·55-s + 4.08e4·57-s − 1.21e4·59-s + 5.10e4·61-s + 1.24e4·65-s + ⋯ |
| L(s) = 1 | + 1.87·3-s + 1.76·5-s + 2.51·9-s − 0.321·11-s + 0.207·13-s + 3.30·15-s − 1.35·17-s + 0.889·19-s − 1.09·23-s + 2.10·25-s + 2.83·27-s + 0.772·29-s + 0.132·31-s − 0.603·33-s − 0.366·37-s + 0.387·39-s + 1.41·41-s − 0.649·43-s + 4.42·45-s + 1.10·47-s − 2.53·51-s − 0.157·53-s − 0.567·55-s + 1.66·57-s − 0.454·59-s + 1.75·61-s + 0.364·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.167352285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.167352285\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 29.2T + 243T^{2} \) |
| 5 | \( 1 - 98.5T + 3.12e3T^{2} \) |
| 11 | \( 1 + 129.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 126.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.61e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.39e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.77e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 710.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.05e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.52e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.67e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.23e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.98e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.68e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.79e4T + 8.58e9T^{2} \) |
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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
−9.446327098272511940481963384753, −8.841354020193449691264253058409, −8.059156668560342844492974968603, −7.02864978359366425129571023888, −6.17179344643558329263562972030, −4.98447086544110312602331464006, −3.86843400481386852952851890279, −2.63894914338290846557212132588, −2.22326553825315008480747927908, −1.24042248246654867813933168934,
1.24042248246654867813933168934, 2.22326553825315008480747927908, 2.63894914338290846557212132588, 3.86843400481386852952851890279, 4.98447086544110312602331464006, 6.17179344643558329263562972030, 7.02864978359366425129571023888, 8.059156668560342844492974968603, 8.841354020193449691264253058409, 9.446327098272511940481963384753
