| L(s) = 1 | + (10 + 5i)5-s − 10i·7-s − 46·11-s + 34i·13-s − 66i·17-s + 104·19-s − 164i·23-s + (75 + 100i)25-s + 224·29-s + 72·31-s + (50 − 100i)35-s − 22i·37-s − 194·41-s + 108i·43-s − 480i·47-s + ⋯ |
| L(s) = 1 | + (0.894 + 0.447i)5-s − 0.539i·7-s − 1.26·11-s + 0.725i·13-s − 0.941i·17-s + 1.25·19-s − 1.48i·23-s + (0.599 + 0.800i)25-s + 1.43·29-s + 0.417·31-s + (0.241 − 0.482i)35-s − 0.0977i·37-s − 0.738·41-s + 0.383i·43-s − 1.48i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.250303144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.250303144\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-10 - 5i)T \) |
| good | 7 | \( 1 + 10iT - 343T^{2} \) |
| 11 | \( 1 + 46T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 104T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 224T + 2.43e4T^{2} \) |
| 31 | \( 1 - 72T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 194T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 480iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 286iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 426T + 2.05e5T^{2} \) |
| 61 | \( 1 - 698T + 2.26e5T^{2} \) |
| 67 | \( 1 + 328iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 188T + 3.57e5T^{2} \) |
| 73 | \( 1 - 740iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 412iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3iT - 9.12e5T^{2} \) |
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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
−10.12930837105072293233241310127, −9.235956296684230092153723229840, −8.206118718911414971480291445458, −7.17889690798557632558515041303, −6.55258745525313880550913429581, −5.38480058724174818601200712536, −4.63413451884388654700570703829, −3.10862309709958647275828695519, −2.27672631087238994064973183352, −0.74034477008234779635989878990,
1.03930901429046610224634402154, 2.31918935324228722664457075427, 3.31968886157543283674587149663, 4.98088694925617918347508684248, 5.47479149004425210354064669062, 6.33137173653031992990131719933, 7.67418672802346128282541758178, 8.323587783727886463857750461091, 9.309979766032039751319327730552, 10.06950187319522645788167453240
