L(s) = 1 | − 17.3·3-s + 25·5-s + 49·7-s + 58.9·9-s + 12.2·11-s − 678.·13-s − 434.·15-s + 1.50e3·17-s − 1.39e3·19-s − 851.·21-s + 4.19e3·23-s + 625·25-s + 3.19e3·27-s + 6.73e3·29-s − 9.61e3·31-s − 213.·33-s + 1.22e3·35-s − 4.55e3·37-s + 1.17e4·39-s + 1.82e4·41-s − 1.84e4·43-s + 1.47e3·45-s − 2.24e4·47-s + 2.40e3·49-s − 2.62e4·51-s − 2.90e4·53-s + 307.·55-s + ⋯ |
L(s) = 1 | − 1.11·3-s + 0.447·5-s + 0.377·7-s + 0.242·9-s + 0.0306·11-s − 1.11·13-s − 0.498·15-s + 1.26·17-s − 0.886·19-s − 0.421·21-s + 1.65·23-s + 0.200·25-s + 0.844·27-s + 1.48·29-s − 1.79·31-s − 0.0341·33-s + 0.169·35-s − 0.546·37-s + 1.24·39-s + 1.69·41-s − 1.52·43-s + 0.108·45-s − 1.48·47-s + 0.142·49-s − 1.41·51-s − 1.42·53-s + 0.0137·55-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 | \( 1 + 17.3T + 243T^{2} \) |
| 11 | \( 1 - 12.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 678.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.50e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.39e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.84e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.24e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.90e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.52e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.73e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.18e5T + 8.58e9T^{2} \) |
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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
−10.65363068317777052560586901972, −9.809792501741617540934767375212, −8.669132243321898517211637360329, −7.40812181585284365082860264846, −6.43155283216311769250923835560, −5.37817334688003719224045933239, −4.74625704202239531172674058320, −2.94438265540481431091055211129, −1.37505942126249257328326042987, 0,
1.37505942126249257328326042987, 2.94438265540481431091055211129, 4.74625704202239531172674058320, 5.37817334688003719224045933239, 6.43155283216311769250923835560, 7.40812181585284365082860264846, 8.669132243321898517211637360329, 9.809792501741617540934767375212, 10.65363068317777052560586901972