| L(s) = 1 | − 2.32·2-s − 1.74·3-s + 3.40·4-s + 4.04·6-s − 3.02·7-s − 3.27·8-s + 0.0282·9-s − 1.65·11-s − 5.93·12-s + 1.59·13-s + 7.03·14-s + 0.804·16-s + 4.50·17-s − 0.0657·18-s − 6.94·19-s + 5.26·21-s + 3.85·22-s − 6.91·23-s + 5.70·24-s − 3.70·26-s + 5.17·27-s − 10.3·28-s − 0.0473·29-s − 31-s + 4.68·32-s + 2.88·33-s − 10.4·34-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 1.00·3-s + 1.70·4-s + 1.65·6-s − 1.14·7-s − 1.15·8-s + 0.00942·9-s − 0.499·11-s − 1.71·12-s + 0.441·13-s + 1.88·14-s + 0.201·16-s + 1.09·17-s − 0.0155·18-s − 1.59·19-s + 1.14·21-s + 0.821·22-s − 1.44·23-s + 1.16·24-s − 0.726·26-s + 0.995·27-s − 1.94·28-s − 0.00878·29-s − 0.179·31-s + 0.828·32-s + 0.501·33-s − 1.79·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2259042061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2259042061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 + 1.74T + 3T^{2} \) |
| 7 | \( 1 + 3.02T + 7T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 13 | \( 1 - 1.59T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 + 6.94T + 19T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 + 0.0473T + 29T^{2} \) |
| 37 | \( 1 + 6.40T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 0.732T + 47T^{2} \) |
| 53 | \( 1 - 2.28T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 4.33T + 61T^{2} \) |
| 67 | \( 1 - 8.81T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 1.80T + 83T^{2} \) |
| 89 | \( 1 - 8.04T + 89T^{2} \) |
| 97 | \( 1 - 3.40T + 97T^{2} \) |
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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
−10.24476202115623028852036249655, −9.669894566454131713329237889817, −8.571524860279930389460331023639, −8.005298887612123777875846132257, −6.74540996789614327602555020833, −6.31658000881862590767945748913, −5.30238116400790841780067956155, −3.63893229891029400078954451125, −2.17300451360371538415838035662, −0.49786466578288976715573408287,
0.49786466578288976715573408287, 2.17300451360371538415838035662, 3.63893229891029400078954451125, 5.30238116400790841780067956155, 6.31658000881862590767945748913, 6.74540996789614327602555020833, 8.005298887612123777875846132257, 8.571524860279930389460331023639, 9.669894566454131713329237889817, 10.24476202115623028852036249655
