| L(s) = 1 | + (−2.22 + 0.184i)5-s + (0.741 + 2.53i)7-s − 2.92i·11-s + 0.587·13-s + 4.81i·17-s + 1.64i·19-s − 2.14·23-s + (4.93 − 0.822i)25-s − 5.05i·29-s + 5.69i·31-s + (−2.12 − 5.52i)35-s − 1.78i·37-s − 7.74·41-s − 4.04i·43-s − 0.204i·47-s + ⋯ |
| L(s) = 1 | + (−0.996 + 0.0825i)5-s + (0.280 + 0.959i)7-s − 0.882i·11-s + 0.163·13-s + 1.16i·17-s + 0.377i·19-s − 0.447·23-s + (0.986 − 0.164i)25-s − 0.938i·29-s + 1.02i·31-s + (−0.358 − 0.933i)35-s − 0.293i·37-s − 1.20·41-s − 0.617i·43-s − 0.0297i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4524036386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4524036386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.22 - 0.184i)T \) |
| 7 | \( 1 + (-0.741 - 2.53i)T \) |
| good | 11 | \( 1 + 2.92iT - 11T^{2} \) |
| 13 | \( 1 - 0.587T + 13T^{2} \) |
| 17 | \( 1 - 4.81iT - 17T^{2} \) |
| 19 | \( 1 - 1.64iT - 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + 5.05iT - 29T^{2} \) |
| 31 | \( 1 - 5.69iT - 31T^{2} \) |
| 37 | \( 1 + 1.78iT - 37T^{2} \) |
| 41 | \( 1 + 7.74T + 41T^{2} \) |
| 43 | \( 1 + 4.04iT - 43T^{2} \) |
| 47 | \( 1 + 0.204iT - 47T^{2} \) |
| 53 | \( 1 - 6.67T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 4.35iT - 61T^{2} \) |
| 67 | \( 1 - 11.7iT - 67T^{2} \) |
| 71 | \( 1 - 3.07iT - 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 9.71T + 79T^{2} \) |
| 83 | \( 1 + 8.45iT - 83T^{2} \) |
| 89 | \( 1 + 6.74T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−8.900446670982316853817378078190, −8.491613911503085068726112996009, −7.965135481646635593741324399737, −7.00381725043052022042007943476, −6.06177071743845157237606481199, −5.49610244014312150702096962456, −4.38366154612257459422867883347, −3.62795117009632162202360715835, −2.74890688937762692318981083932, −1.48380382579194106142842741147,
0.15656873372744689461191254176, 1.44212836681647231193640770020, 2.84512027805530542305128622065, 3.80769258370635183085023963376, 4.55916315685638967869063923679, 5.11811677105467958259005330090, 6.49740600619031307741189504493, 7.22017894345406642777022110686, 7.64355230933243337734789909996, 8.436439346331251102424659123628
