L(s) = 1 | − 53.1·7-s − 585.·11-s + 1.08e3·13-s − 1.59e3·17-s + 2.41e3·19-s + 2.67e3·23-s − 8.43e3·29-s − 5.02e3·31-s − 1.82e3·37-s − 7.97e3·41-s − 1.05e4·43-s + 2.93e4·47-s − 1.39e4·49-s + 2.30e4·53-s + 2.21e3·59-s + 2.19e4·61-s + 410.·67-s − 5.36e4·71-s + 1.75e4·73-s + 3.11e4·77-s + 5.95e4·79-s − 2.29e4·83-s − 5.67e4·89-s − 5.76e4·91-s + 1.51e5·97-s + 1.13e5·101-s − 1.67e4·103-s + ⋯ |
L(s) = 1 | − 0.409·7-s − 1.45·11-s + 1.78·13-s − 1.33·17-s + 1.53·19-s + 1.05·23-s − 1.86·29-s − 0.938·31-s − 0.219·37-s − 0.740·41-s − 0.870·43-s + 1.94·47-s − 0.831·49-s + 1.12·53-s + 0.0828·59-s + 0.754·61-s + 0.0111·67-s − 1.26·71-s + 0.385·73-s + 0.598·77-s + 1.07·79-s − 0.365·83-s − 0.758·89-s − 0.730·91-s + 1.63·97-s + 1.10·101-s − 0.155·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.674626907\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.674626907\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 53.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 585.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.08e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.59e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.41e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.97e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.93e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.30e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.21e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 410.T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.36e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.51e5T + 8.58e9T^{2} \) |
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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
−9.215984433318627317029692165431, −8.668813216638605461622582178491, −7.61593758083895364854681572338, −6.89437410713311505861737811028, −5.77693360549079219782887842571, −5.16967567858661910001126584212, −3.83481573714663695034983470021, −3.05275297226028945463651969622, −1.84737848802021206161363289703, −0.56870115973640364230081451196,
0.56870115973640364230081451196, 1.84737848802021206161363289703, 3.05275297226028945463651969622, 3.83481573714663695034983470021, 5.16967567858661910001126584212, 5.77693360549079219782887842571, 6.89437410713311505861737811028, 7.61593758083895364854681572338, 8.668813216638605461622582178491, 9.215984433318627317029692165431