| L(s) = 1 | − 2-s + 4-s + (1.60 + 1.55i)5-s − 8-s + (−1.60 − 1.55i)10-s − 4.10i·11-s − 2.67·13-s + 16-s + 1.29i·17-s − 6.96i·19-s + (1.60 + 1.55i)20-s + 4.10i·22-s + 3.53·23-s + (0.143 + 4.99i)25-s + 2.67·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.717 + 0.696i)5-s − 0.353·8-s + (−0.507 − 0.492i)10-s − 1.23i·11-s − 0.742·13-s + 0.250·16-s + 0.314i·17-s − 1.59i·19-s + (0.358 + 0.348i)20-s + 0.875i·22-s + 0.736·23-s + (0.0287 + 0.999i)25-s + 0.525·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.471929335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.471929335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.60 - 1.55i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.10iT - 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 1.29iT - 17T^{2} \) |
| 19 | \( 1 + 6.96iT - 19T^{2} \) |
| 23 | \( 1 - 3.53T + 23T^{2} \) |
| 29 | \( 1 - 3.22iT - 29T^{2} \) |
| 31 | \( 1 - 8.38iT - 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 + 1.99T + 41T^{2} \) |
| 43 | \( 1 - 0.0984iT - 43T^{2} \) |
| 47 | \( 1 + 9.68iT - 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 0.796T + 59T^{2} \) |
| 61 | \( 1 + 2.69iT - 61T^{2} \) |
| 67 | \( 1 + 0.696iT - 67T^{2} \) |
| 71 | \( 1 + 9.32iT - 71T^{2} \) |
| 73 | \( 1 - 7.26T + 73T^{2} \) |
| 79 | \( 1 - 2.73T + 79T^{2} \) |
| 83 | \( 1 + 6.79iT - 83T^{2} \) |
| 89 | \( 1 - 8.03T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−8.692995612097898814084547038044, −7.58135882304146161405720062437, −6.78924951284737636461092464462, −6.53099302460231363684170473977, −5.44345743691827608566859843522, −4.88519957165827323350614365887, −3.33334602321746910809269394616, −2.91238978742710754660163142403, −1.90523276472273828408991331432, −0.72023630307867517304345389391,
0.792258297981659850513619536513, 1.96334977682017750976748730474, 2.42836475928977854222412598597, 3.88221835634191403182824208030, 4.67325197424601180135215206850, 5.55131000876376434112954872400, 6.11506159658401463218859605723, 7.17228620751595941925320106188, 7.61195958208665532357045895701, 8.404178860901200054562331964211
