| L(s) = 1 | − 3.13·3-s − 3.00·5-s − 0.714·7-s + 6.82·9-s + 1.17·11-s + 4.39·13-s + 9.40·15-s − 2.25·17-s + 8.02·19-s + 2.23·21-s + 4.00·25-s − 11.9·27-s − 8.34·29-s + 3.19·31-s − 3.69·33-s + 2.14·35-s + 4.74·37-s − 13.7·39-s − 10.8·41-s + 8.08·43-s − 20.4·45-s + 1.20·47-s − 6.48·49-s + 7.05·51-s + 0.460·53-s − 3.53·55-s − 25.1·57-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 1.34·5-s − 0.269·7-s + 2.27·9-s + 0.354·11-s + 1.21·13-s + 2.42·15-s − 0.545·17-s + 1.84·19-s + 0.488·21-s + 0.801·25-s − 2.30·27-s − 1.54·29-s + 0.574·31-s − 0.642·33-s + 0.362·35-s + 0.780·37-s − 2.20·39-s − 1.70·41-s + 1.23·43-s − 3.05·45-s + 0.175·47-s − 0.927·49-s + 0.987·51-s + 0.0632·53-s − 0.476·55-s − 3.33·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6195835396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6195835396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 5 | \( 1 + 3.00T + 5T^{2} \) |
| 7 | \( 1 + 0.714T + 7T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 - 4.39T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 8.02T + 19T^{2} \) |
| 29 | \( 1 + 8.34T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8.08T + 43T^{2} \) |
| 47 | \( 1 - 1.20T + 47T^{2} \) |
| 53 | \( 1 - 0.460T + 53T^{2} \) |
| 59 | \( 1 - 4.53T + 59T^{2} \) |
| 61 | \( 1 - 2.56T + 61T^{2} \) |
| 67 | \( 1 + 3.89T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 + 0.0978T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 1.88T + 83T^{2} \) |
| 89 | \( 1 + 5.90T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.230465746651060397417090228866, −7.40314731371507788324230486898, −6.95070000939671041781450351982, −6.09697297956988938672980459442, −5.55270421557237345778490078306, −4.66692611008277951116163541609, −3.97805123766032380710836602432, −3.30478839243943206159918257903, −1.43716977819071094388698681332, −0.53594425065794722681186413665,
0.53594425065794722681186413665, 1.43716977819071094388698681332, 3.30478839243943206159918257903, 3.97805123766032380710836602432, 4.66692611008277951116163541609, 5.55270421557237345778490078306, 6.09697297956988938672980459442, 6.95070000939671041781450351982, 7.40314731371507788324230486898, 8.230465746651060397417090228866
