| L(s) = 1 | + i·5-s − 1.88i·7-s + 5.94i·11-s + (−1.74 + 3.15i)13-s − 1.88·17-s + 0.474i·19-s − 4.42·23-s − 25-s − 10.5·29-s − 6.30i·31-s + 1.88·35-s − 3.94i·37-s − 7.94i·41-s + 3.44·49-s + 8.66·53-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.712i·7-s + 1.79i·11-s + (−0.484 + 0.874i)13-s − 0.457·17-s + 0.108i·19-s − 0.922·23-s − 0.200·25-s − 1.95·29-s − 1.13i·31-s + 0.318·35-s − 0.648i·37-s − 1.24i·41-s + 0.492·49-s + 1.19·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4004460264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4004460264\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 13 | \( 1 + (1.74 - 3.15i)T \) |
| good | 7 | \( 1 + 1.88iT - 7T^{2} \) |
| 11 | \( 1 - 5.94iT - 11T^{2} \) |
| 17 | \( 1 + 1.88T + 17T^{2} \) |
| 19 | \( 1 - 0.474iT - 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 6.30iT - 31T^{2} \) |
| 37 | \( 1 + 3.94iT - 37T^{2} \) |
| 41 | \( 1 + 7.94iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 2.44T + 61T^{2} \) |
| 67 | \( 1 + 11.0iT - 67T^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 + 13.0iT - 73T^{2} \) |
| 79 | \( 1 + 3.55T + 79T^{2} \) |
| 83 | \( 1 - 7.88iT - 83T^{2} \) |
| 89 | \( 1 - 0.0563iT - 89T^{2} \) |
| 97 | \( 1 + 18.2iT - 97T^{2} \) |
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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
−7.71474618817321687586435156151, −7.35155694759337386465550100383, −6.84475134227235538260807087087, −5.95056074729125530324031956299, −5.01248598024632311107157229693, −4.12667317233554335864395282066, −3.81611747064087299854360997061, −2.21218141185341616810223978006, −1.93799960981962002897309106001, −0.11248940894566140667745005937,
1.09212382097852168790241993484, 2.34936154333439960773364833057, 3.15849168675603524004053867919, 3.95668152889688755436556083292, 5.04359752728764659306485615623, 5.69189985294580385477179120760, 6.08459081950568528020290037308, 7.17574218769840423173105798063, 7.984091593542024127707615512005, 8.613345493281897105772087583094
