| L(s) = 1 | + 2-s + 4-s − 3.16·5-s − 7-s + 8-s − 3.16·10-s − 11-s + 2·13-s − 14-s + 16-s + 7.16·17-s + 5.16·19-s − 3.16·20-s − 22-s − 6.32·23-s + 5.00·25-s + 2·26-s − 28-s + 4·29-s + 9.16·31-s + 32-s + 7.16·34-s + 3.16·35-s + 8.32·37-s + 5.16·38-s − 3.16·40-s − 7.16·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.41·5-s − 0.377·7-s + 0.353·8-s − 1.00·10-s − 0.301·11-s + 0.554·13-s − 0.267·14-s + 0.250·16-s + 1.73·17-s + 1.18·19-s − 0.707·20-s − 0.213·22-s − 1.31·23-s + 1.00·25-s + 0.392·26-s − 0.188·28-s + 0.742·29-s + 1.64·31-s + 0.176·32-s + 1.22·34-s + 0.534·35-s + 1.36·37-s + 0.837·38-s − 0.500·40-s − 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.035207857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.035207857\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3.16T + 5T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.16T + 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 - 8.32T + 37T^{2} \) |
| 41 | \( 1 + 7.16T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 4.32T + 53T^{2} \) |
| 59 | \( 1 + 1.67T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 + 4.32T + 79T^{2} \) |
| 83 | \( 1 + 5.16T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−9.848443317916173747335658826716, −8.417730207233797965216599747173, −7.898183885753629869464442637151, −7.21738729521407466383350349533, −6.18375882364950955616071480329, −5.35817656475985994720849351478, −4.30571643462170437746204485552, −3.57050061363473802858282797451, −2.84079256267322368630215735514, −0.961730809401341751806754824200,
0.961730809401341751806754824200, 2.84079256267322368630215735514, 3.57050061363473802858282797451, 4.30571643462170437746204485552, 5.35817656475985994720849351478, 6.18375882364950955616071480329, 7.21738729521407466383350349533, 7.898183885753629869464442637151, 8.417730207233797965216599747173, 9.848443317916173747335658826716
