L(s) = 1 | − 21.3·3-s − 25·5-s + 49·7-s + 211.·9-s − 381.·11-s + 372.·13-s + 533.·15-s − 1.09e3·17-s + 2.85e3·19-s − 1.04e3·21-s + 2.94e3·23-s + 625·25-s + 666.·27-s + 2.41e3·29-s − 463.·31-s + 8.13e3·33-s − 1.22e3·35-s + 1.80e3·37-s − 7.93e3·39-s − 9.01e3·41-s − 33.7·43-s − 5.29e3·45-s − 1.51e4·47-s + 2.40e3·49-s + 2.34e4·51-s − 3.13e4·53-s + 9.53e3·55-s + ⋯ |
L(s) = 1 | − 1.36·3-s − 0.447·5-s + 0.377·7-s + 0.871·9-s − 0.950·11-s + 0.610·13-s + 0.611·15-s − 0.922·17-s + 1.81·19-s − 0.517·21-s + 1.16·23-s + 0.200·25-s + 0.176·27-s + 0.532·29-s − 0.0866·31-s + 1.30·33-s − 0.169·35-s + 0.216·37-s − 0.835·39-s − 0.837·41-s − 0.00278·43-s − 0.389·45-s − 0.999·47-s + 0.142·49-s + 1.26·51-s − 1.53·53-s + 0.425·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 - 49T \) |
good | 3 | \( 1 + 21.3T + 243T^{2} \) |
| 11 | \( 1 + 381.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 372.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.85e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.94e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 463.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 33.7T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.13e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.28e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.76e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.82e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.32e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.80e4T + 8.58e9T^{2} \) |
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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
−11.00969926878326314633067705147, −9.815722166978925485791940218576, −8.522591873219258337790262324755, −7.45667061099067271546080410523, −6.47228051255136927213197238756, −5.33314083957624770944359208082, −4.70986619520904121934010860743, −3.09685932900926104405670822259, −1.19399614153027410820455406148, 0,
1.19399614153027410820455406148, 3.09685932900926104405670822259, 4.70986619520904121934010860743, 5.33314083957624770944359208082, 6.47228051255136927213197238756, 7.45667061099067271546080410523, 8.522591873219258337790262324755, 9.815722166978925485791940218576, 11.00969926878326314633067705147