| L(s) = 1 | − 3-s + 4.19·5-s − 1.43·7-s + 9-s − 5.55·11-s − 2.46·13-s − 4.19·15-s − 0.151·17-s − 0.370·19-s + 1.43·21-s + 5.22·23-s + 12.6·25-s − 27-s − 9.43·29-s − 6.03·31-s + 5.55·33-s − 6.01·35-s + 9.91·37-s + 2.46·39-s − 8.03·41-s + 12.3·43-s + 4.19·45-s + 5.19·47-s − 4.94·49-s + 0.151·51-s + 6.02·53-s − 23.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.87·5-s − 0.541·7-s + 0.333·9-s − 1.67·11-s − 0.683·13-s − 1.08·15-s − 0.0368·17-s − 0.0849·19-s + 0.312·21-s + 1.08·23-s + 2.52·25-s − 0.192·27-s − 1.75·29-s − 1.08·31-s + 0.966·33-s − 1.01·35-s + 1.62·37-s + 0.394·39-s − 1.25·41-s + 1.88·43-s + 0.625·45-s + 0.757·47-s − 0.706·49-s + 0.0212·51-s + 0.827·53-s − 3.14·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.751571692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751571692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 4.19T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 5.55T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 0.151T + 17T^{2} \) |
| 19 | \( 1 + 0.370T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 9.43T + 29T^{2} \) |
| 31 | \( 1 + 6.03T + 31T^{2} \) |
| 37 | \( 1 - 9.91T + 37T^{2} \) |
| 41 | \( 1 + 8.03T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 - 6.02T + 53T^{2} \) |
| 59 | \( 1 - 6.13T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 - 6.05T + 67T^{2} \) |
| 71 | \( 1 + 3.38T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 5.34T + 79T^{2} \) |
| 83 | \( 1 + 1.83T + 83T^{2} \) |
| 89 | \( 1 - 7.69T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−7.45966877260441634478995547307, −7.24058034876925340233788451633, −6.17168216956088652114120082493, −5.75994230358819945101196904218, −5.24447766351789670591487747243, −4.64148103102347071241035434042, −3.29731819657770232180853934534, −2.46324807194289657055643983702, −1.95857005234615926869685949532, −0.64893896729074391839953749161,
0.64893896729074391839953749161, 1.95857005234615926869685949532, 2.46324807194289657055643983702, 3.29731819657770232180853934534, 4.64148103102347071241035434042, 5.24447766351789670591487747243, 5.75994230358819945101196904218, 6.17168216956088652114120082493, 7.24058034876925340233788451633, 7.45966877260441634478995547307
