| L(s) = 1 | + 2·2-s + (−1.06 + 5.08i)3-s + 4·4-s − 5i·5-s + (−2.13 + 10.1i)6-s + 2.56·7-s + 8·8-s + (−24.7 − 10.8i)9-s − 10i·10-s − 12.9i·11-s + (−4.27 + 20.3i)12-s − 54.6i·13-s + 5.13·14-s + (25.4 + 5.34i)15-s + 16·16-s + 33.9i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (−0.205 + 0.978i)3-s + 0.5·4-s − 0.447i·5-s + (−0.145 + 0.691i)6-s + 0.138·7-s + 0.353·8-s + (−0.915 − 0.403i)9-s − 0.316i·10-s − 0.355i·11-s + (−0.102 + 0.489i)12-s − 1.16i·13-s + 0.0980·14-s + (0.437 + 0.0920i)15-s + 0.250·16-s + 0.483i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.045923451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045923451\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + (1.06 - 5.08i)T \) |
| 5 | \( 1 + 5iT \) |
| 19 | \( 1 + (49.6 + 66.3i)T \) |
| good | 7 | \( 1 - 2.56T + 343T^{2} \) |
| 11 | \( 1 + 12.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 54.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 33.9iT - 4.91e3T^{2} \) |
| 23 | \( 1 + 21.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 249. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 206. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 372.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 73.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 32.6iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 289.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 352.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 518.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 772. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 448.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 911. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 197.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3iT - 9.12e5T^{2} \) |
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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
−10.37740235264072800320537689426, −9.364215772453750282093394374414, −8.495361329134821548984281163462, −7.50507690079469385316901343944, −6.01985683817834057836982738304, −5.51967719098330441810493129185, −4.46251947999588786468068741728, −3.67512553404290522430167581017, −2.45055058475564327328948540573, −0.47241795372785000063583169433,
1.51253063819032418824671785338, 2.46735830696910658108442456876, 3.78412668365442858226668300309, 5.00134450810299152639608505188, 6.03774104886340376271781079818, 6.83693675347202698470555286169, 7.47285750823132586299810009914, 8.525498287547132209961648861310, 9.685341051011369402064640109755, 10.88123529874279270840503339348
