| L(s) = 1 | + 1.61i·2-s − 0.618·4-s − i·7-s + 2.23i·8-s − 4.23·11-s − 3.23i·13-s + 1.61·14-s − 4.85·16-s − 6.47i·17-s − 4.47·19-s − 6.85i·22-s − 1.76i·23-s + 5.23·26-s + 0.618i·28-s + 5·29-s + ⋯ |
| L(s) = 1 | + 1.14i·2-s − 0.309·4-s − 0.377i·7-s + 0.790i·8-s − 1.27·11-s − 0.897i·13-s + 0.432·14-s − 1.21·16-s − 1.56i·17-s − 1.02·19-s − 1.46i·22-s − 0.367i·23-s + 1.02·26-s + 0.116i·28-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5858366119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5858366119\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 2 | \( 1 - 1.61iT - 2T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 1.76iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 + 9.23T + 41T^{2} \) |
| 43 | \( 1 - 6.23iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 0.236iT - 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 + 13.2iT - 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5.70iT - 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 0.763iT - 97T^{2} \) |
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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
−8.991439304762576948147607436957, −8.237441655284061715357283553395, −7.57474943540376949801708080009, −7.00850355655284095450038166823, −6.09512586848628414570875809501, −5.23017111609629045380200451711, −4.71282426522890702559919554163, −3.18946847508676056083301352717, −2.24194350641245963545125766032, −0.20466157428546507995130571167,
1.67129052853386935293489699017, 2.34242110252294544048449467112, 3.44687450093396058924399961979, 4.25760384400125752435841112737, 5.34087613410524671933726164129, 6.33541183611827990986069855168, 7.11731392585546893078341336906, 8.244897177322079558394501989112, 8.845489377448292340038943045536, 9.858996289499288182833782231531
