| L(s) = 1 | + 3-s + 2·7-s + 9-s + 4·11-s − 13-s − 17-s − 2·19-s + 2·21-s − 7·23-s + 27-s − 6·29-s − 4·31-s + 4·33-s − 11·37-s − 39-s − 9·41-s + 6·43-s + 4·47-s − 3·49-s − 51-s + 53-s − 2·57-s + 59-s + 5·61-s + 2·63-s − 2·67-s − 7·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.242·17-s − 0.458·19-s + 0.436·21-s − 1.45·23-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s − 1.80·37-s − 0.160·39-s − 1.40·41-s + 0.914·43-s + 0.583·47-s − 3/7·49-s − 0.140·51-s + 0.137·53-s − 0.264·57-s + 0.130·59-s + 0.640·61-s + 0.251·63-s − 0.244·67-s − 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.678261368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.678261368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.64297896725778, −13.00292601903360, −12.46772891677225, −11.97080186168297, −11.66664684440357, −11.08045188738194, −10.50796146551535, −10.13146549203238, −9.399572682801877, −9.143166930576930, −8.481181730145630, −8.283431657130921, −7.442810957372846, −7.249581308279708, −6.532193656808347, −6.064014633336894, −5.374876258822189, −4.872723414214845, −4.173727598347491, −3.790358739333412, −3.349178900685887, −2.361455678080438, −1.826840627954385, −1.556193915732792, −0.4474160365710492,
0.4474160365710492, 1.556193915732792, 1.826840627954385, 2.361455678080438, 3.349178900685887, 3.790358739333412, 4.173727598347491, 4.872723414214845, 5.374876258822189, 6.064014633336894, 6.532193656808347, 7.249581308279708, 7.442810957372846, 8.283431657130921, 8.481181730145630, 9.143166930576930, 9.399572682801877, 10.13146549203238, 10.50796146551535, 11.08045188738194, 11.66664684440357, 11.97080186168297, 12.46772891677225, 13.00292601903360, 13.64297896725778
