| L(s) = 1 | − 0.899·2-s − 0.0342·3-s − 1.19·4-s − 5-s + 0.0307·6-s + 4.96·7-s + 2.86·8-s − 2.99·9-s + 0.899·10-s − 4.48·11-s + 0.0407·12-s + 4.29·13-s − 4.46·14-s + 0.0342·15-s − 0.196·16-s − 6.85·17-s + 2.69·18-s + 4.86·19-s + 1.19·20-s − 0.170·21-s + 4.03·22-s + 7.38·23-s − 0.0981·24-s + 25-s − 3.85·26-s + 0.205·27-s − 5.92·28-s + ⋯ |
| L(s) = 1 | − 0.635·2-s − 0.0197·3-s − 0.595·4-s − 0.447·5-s + 0.0125·6-s + 1.87·7-s + 1.01·8-s − 0.999·9-s + 0.284·10-s − 1.35·11-s + 0.0117·12-s + 1.18·13-s − 1.19·14-s + 0.00883·15-s − 0.0490·16-s − 1.66·17-s + 0.635·18-s + 1.11·19-s + 0.266·20-s − 0.0371·21-s + 0.859·22-s + 1.53·23-s − 0.0200·24-s + 0.200·25-s − 0.756·26-s + 0.0394·27-s − 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9237325673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9237325673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 0.899T + 2T^{2} \) |
| 3 | \( 1 + 0.0342T + 3T^{2} \) |
| 7 | \( 1 - 4.96T + 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 + 6.85T + 17T^{2} \) |
| 19 | \( 1 - 4.86T + 19T^{2} \) |
| 23 | \( 1 - 7.38T + 23T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 + 0.591T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 8.46T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 5.62T + 61T^{2} \) |
| 67 | \( 1 + 2.54T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 + 8.42T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 5.34T + 89T^{2} \) |
| 97 | \( 1 + 0.129T + 97T^{2} \) |
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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
−10.66270983180910586365919909550, −9.025128677959796006571030753613, −8.748417604523737099812185345316, −7.895427218840449972076836687014, −7.39111442385679109655513348542, −5.61267737980436523266443508526, −4.98537799544368685081112126547, −4.06041655961473350929381549695, −2.46018481839399139384085754564, −0.913824776526714683837526822830,
0.913824776526714683837526822830, 2.46018481839399139384085754564, 4.06041655961473350929381549695, 4.98537799544368685081112126547, 5.61267737980436523266443508526, 7.39111442385679109655513348542, 7.895427218840449972076836687014, 8.748417604523737099812185345316, 9.025128677959796006571030753613, 10.66270983180910586365919909550
