| L(s) = 1 | − 0.414·2-s − 1.82·4-s + 2.82·5-s + 2.82·7-s + 1.58·8-s − 1.17·10-s + 2·11-s − 13-s − 1.17·14-s + 3·16-s − 7.65·17-s − 2.82·19-s − 5.17·20-s − 0.828·22-s + 4·23-s + 3.00·25-s + 0.414·26-s − 5.17·28-s − 2·29-s − 1.17·31-s − 4.41·32-s + 3.17·34-s + 8.00·35-s − 7.65·37-s + 1.17·38-s + 4.48·40-s − 5.17·41-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s + 1.26·5-s + 1.06·7-s + 0.560·8-s − 0.370·10-s + 0.603·11-s − 0.277·13-s − 0.313·14-s + 0.750·16-s − 1.85·17-s − 0.648·19-s − 1.15·20-s − 0.176·22-s + 0.834·23-s + 0.600·25-s + 0.0812·26-s − 0.977·28-s − 0.371·29-s − 0.210·31-s − 0.780·32-s + 0.543·34-s + 1.35·35-s − 1.25·37-s + 0.190·38-s + 0.709·40-s − 0.807·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9790522203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9790522203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 3.65T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−13.64092450625649755112962206173, −12.77900343982831049892648522426, −11.25744650776750850906437862654, −10.27099661688147708313866214628, −9.134631821518355283854518142595, −8.549601039417608882680339987542, −6.90878693007941002147044931140, −5.42948244887922082375027507222, −4.37656920736378168473585285560, −1.87345696301588889971226093835,
1.87345696301588889971226093835, 4.37656920736378168473585285560, 5.42948244887922082375027507222, 6.90878693007941002147044931140, 8.549601039417608882680339987542, 9.134631821518355283854518142595, 10.27099661688147708313866214628, 11.25744650776750850906437862654, 12.77900343982831049892648522426, 13.64092450625649755112962206173
