| L(s) = 1 | + 1.76·3-s − 3.11·5-s + 7-s + 0.108·9-s + 5.23·11-s + 6.34·13-s − 5.49·15-s − 0.585·17-s − 8.48·19-s + 1.76·21-s + 23-s + 4.71·25-s − 5.09·27-s − 1.59·29-s + 5.97·31-s + 9.23·33-s − 3.11·35-s + 10.0·37-s + 11.1·39-s + 1.17·41-s + 2.96·43-s − 0.339·45-s + 2.55·47-s + 49-s − 1.03·51-s + 8.27·53-s − 16.3·55-s + ⋯ |
| L(s) = 1 | + 1.01·3-s − 1.39·5-s + 0.377·7-s + 0.0363·9-s + 1.57·11-s + 1.76·13-s − 1.41·15-s − 0.141·17-s − 1.94·19-s + 0.384·21-s + 0.208·23-s + 0.942·25-s − 0.981·27-s − 0.296·29-s + 1.07·31-s + 1.60·33-s − 0.526·35-s + 1.64·37-s + 1.79·39-s + 0.183·41-s + 0.451·43-s − 0.0505·45-s + 0.372·47-s + 0.142·49-s − 0.144·51-s + 1.13·53-s − 2.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.310328268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.310328268\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 + 0.585T + 17T^{2} \) |
| 19 | \( 1 + 8.48T + 19T^{2} \) |
| 29 | \( 1 + 1.59T + 29T^{2} \) |
| 31 | \( 1 - 5.97T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 - 2.96T + 43T^{2} \) |
| 47 | \( 1 - 2.55T + 47T^{2} \) |
| 53 | \( 1 - 8.27T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 - 6.15T + 61T^{2} \) |
| 67 | \( 1 - 8.52T + 67T^{2} \) |
| 71 | \( 1 + 8.38T + 71T^{2} \) |
| 73 | \( 1 + 0.358T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 8.38T + 97T^{2} \) |
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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
−8.882803426037346931589717861685, −8.172986363615837302882697280586, −7.71418742173140840015486469266, −6.56401275347931671215129980264, −6.08788343655632079085084794003, −4.43064318453531092451110621213, −3.99199321316709254929721944449, −3.42026975294959924393039784737, −2.20971107103952042064359056944, −0.953993504108400945934873769286,
0.953993504108400945934873769286, 2.20971107103952042064359056944, 3.42026975294959924393039784737, 3.99199321316709254929721944449, 4.43064318453531092451110621213, 6.08788343655632079085084794003, 6.56401275347931671215129980264, 7.71418742173140840015486469266, 8.172986363615837302882697280586, 8.882803426037346931589717861685
