| L(s) = 1 | + (0.618 + 1.61i)3-s + (−2.61 + 0.381i)7-s + (−2.23 + 2.00i)9-s − 0.763i·11-s + 1.23·13-s − 4.47i·17-s + 7.23i·19-s + (−2.23 − 4i)21-s − 7.70·23-s + (−4.61 − 2.38i)27-s − 4i·29-s + 3.23i·31-s + (1.23 − 0.472i)33-s − 4.47i·37-s + (0.763 + 2.00i)39-s + ⋯ |
| L(s) = 1 | + (0.356 + 0.934i)3-s + (−0.989 + 0.144i)7-s + (−0.745 + 0.666i)9-s − 0.230i·11-s + 0.342·13-s − 1.08i·17-s + 1.66i·19-s + (−0.487 − 0.872i)21-s − 1.60·23-s + (−0.888 − 0.458i)27-s − 0.742i·29-s + 0.581i·31-s + (0.215 − 0.0821i)33-s − 0.735i·37-s + (0.122 + 0.320i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 - 1.61i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.61 - 0.381i)T \) |
| good | 11 | \( 1 + 0.763iT - 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 7.23iT - 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 3.23iT - 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 7.23iT - 43T^{2} \) |
| 47 | \( 1 - 3.23iT - 47T^{2} \) |
| 53 | \( 1 + 9.23T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 4.94iT - 61T^{2} \) |
| 67 | \( 1 + 9.70iT - 67T^{2} \) |
| 71 | \( 1 + 12.1iT - 71T^{2} \) |
| 73 | \( 1 + 6.76T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.23iT - 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−9.003395703471079924711898143030, −8.159354523411582557062867095870, −7.46195470157238639818769536690, −6.17708619993648500037950859593, −5.78757460471766174981593649916, −4.67200274374457960873776453965, −3.73494570595563733403600406831, −3.18289205287173802321553842123, −2.03020804236914376588541850359, 0,
1.45136778441717679696762422304, 2.57943264780482616975443102511, 3.41368988455271101343075851536, 4.37580328612576462909410838030, 5.70165169085963980366183265607, 6.42208878114882333965518959716, 6.91254053070562514935503473736, 7.84718156724336612229851981255, 8.508993516215303599650888680214
