| L(s) = 1 | − 3.44·2-s + 3.86·4-s + 20.9·5-s − 30.0·7-s + 14.2·8-s − 72.0·10-s − 51.2·11-s − 22.3·13-s + 103.·14-s − 79.9·16-s + 89.1·17-s + 96.6·19-s + 80.7·20-s + 176.·22-s − 76.1·23-s + 312.·25-s + 76.8·26-s − 116.·28-s + 71.5·29-s − 129.·31-s + 161.·32-s − 307.·34-s − 628.·35-s − 108.·37-s − 332.·38-s + 297.·40-s + 357.·41-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 0.482·4-s + 1.86·5-s − 1.62·7-s + 0.629·8-s − 2.27·10-s − 1.40·11-s − 0.475·13-s + 1.97·14-s − 1.24·16-s + 1.27·17-s + 1.16·19-s + 0.902·20-s + 1.70·22-s − 0.690·23-s + 2.49·25-s + 0.579·26-s − 0.783·28-s + 0.458·29-s − 0.749·31-s + 0.892·32-s − 1.54·34-s − 3.03·35-s − 0.481·37-s − 1.42·38-s + 1.17·40-s + 1.36·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9711976148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9711976148\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 + 3.44T + 8T^{2} \) |
| 5 | \( 1 - 20.9T + 125T^{2} \) |
| 7 | \( 1 + 30.0T + 343T^{2} \) |
| 11 | \( 1 + 51.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 89.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 96.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 71.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 108.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 357.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 237.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 97.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 705.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 549.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 652.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 37.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 577.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 375.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 322.T + 9.12e5T^{2} \) |
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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
−10.09588479976130059790073245371, −9.684865951740472573414423384928, −9.068760585529539240793770619125, −7.81993895747566955440739863557, −6.94690340755505442338000357531, −5.86867955725547276401745380033, −5.19375934598890235180234179354, −3.12920107123027876446619366250, −2.14476389175076878253410387962, −0.71342768467680095642714201685,
0.71342768467680095642714201685, 2.14476389175076878253410387962, 3.12920107123027876446619366250, 5.19375934598890235180234179354, 5.86867955725547276401745380033, 6.94690340755505442338000357531, 7.81993895747566955440739863557, 9.068760585529539240793770619125, 9.684865951740472573414423384928, 10.09588479976130059790073245371
