| L(s) = 1 | + 2.96·5-s + 3.35·7-s + 1.61·11-s + 13-s − 2·17-s − 3.35·19-s + 6.70·23-s + 3.77·25-s − 2·29-s + 6.57·31-s + 9.92·35-s + 7.92·37-s − 6.96·41-s − 0.775·43-s − 2.38·47-s + 4.22·49-s − 11.9·53-s + 4.77·55-s − 0.312·59-s + 14.6·61-s + 2.96·65-s + 8.12·67-s + 4.31·71-s + 0.0752·73-s + 5.40·77-s + 12·79-s + 8.31·83-s + ⋯ |
| L(s) = 1 | + 1.32·5-s + 1.26·7-s + 0.486·11-s + 0.277·13-s − 0.485·17-s − 0.768·19-s + 1.39·23-s + 0.755·25-s − 0.371·29-s + 1.18·31-s + 1.67·35-s + 1.30·37-s − 1.08·41-s − 0.118·43-s − 0.348·47-s + 0.603·49-s − 1.63·53-s + 0.643·55-s − 0.0407·59-s + 1.87·61-s + 0.367·65-s + 0.992·67-s + 0.511·71-s + 0.00880·73-s + 0.615·77-s + 1.35·79-s + 0.912·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.133239875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.133239875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.96T + 5T^{2} \) |
| 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.57T + 31T^{2} \) |
| 37 | \( 1 - 7.92T + 37T^{2} \) |
| 41 | \( 1 + 6.96T + 41T^{2} \) |
| 43 | \( 1 + 0.775T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 0.312T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 8.12T + 67T^{2} \) |
| 71 | \( 1 - 4.31T + 71T^{2} \) |
| 73 | \( 1 - 0.0752T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 8.31T + 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 + 7.92T + 97T^{2} \) |
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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
−8.447747150468574211893470506133, −8.009824560820603507944535695234, −6.75304339369194624879972398698, −6.43164197649586660337400913161, −5.39699733556648545286361475261, −4.89055907670581697658686369348, −4.02835875087941185941142377188, −2.73834803168262196239306174899, −1.91220421895517642299308082562, −1.14540152506832449270669972735,
1.14540152506832449270669972735, 1.91220421895517642299308082562, 2.73834803168262196239306174899, 4.02835875087941185941142377188, 4.89055907670581697658686369348, 5.39699733556648545286361475261, 6.43164197649586660337400913161, 6.75304339369194624879972398698, 8.009824560820603507944535695234, 8.447747150468574211893470506133
