| L(s) = 1 | − 1.77·2-s + 1.14·4-s − 0.589·5-s − 7-s + 1.52·8-s + 1.04·10-s − 1.54·13-s + 1.77·14-s − 4.98·16-s − 7.14·17-s − 6.19·19-s − 0.672·20-s − 3.76·23-s − 4.65·25-s + 2.73·26-s − 1.14·28-s − 0.607·29-s − 6.87·31-s + 5.78·32-s + 12.6·34-s + 0.589·35-s − 7.70·37-s + 10.9·38-s − 0.898·40-s − 7.48·41-s − 10.9·43-s + 6.66·46-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.570·4-s − 0.263·5-s − 0.377·7-s + 0.538·8-s + 0.330·10-s − 0.428·13-s + 0.473·14-s − 1.24·16-s − 1.73·17-s − 1.42·19-s − 0.150·20-s − 0.784·23-s − 0.930·25-s + 0.536·26-s − 0.215·28-s − 0.112·29-s − 1.23·31-s + 1.02·32-s + 2.17·34-s + 0.0997·35-s − 1.26·37-s + 1.77·38-s − 0.142·40-s − 1.16·41-s − 1.67·43-s + 0.983·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05217521080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05217521080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 5 | \( 1 + 0.589T + 5T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 + 7.14T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 0.607T + 29T^{2} \) |
| 31 | \( 1 + 6.87T + 31T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 7.04T + 53T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 + 2.19T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 2.58T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 + 8.15T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−8.115219961734283907776990007599, −7.20632461540460542280761628684, −6.80932447300415709737439928025, −6.04781955065939962990260466204, −4.98145696792792449552037861277, −4.27440674080925693175285995435, −3.56959499915042015431233962641, −2.19599488161042912285673705934, −1.82410748104880055423200393360, −0.13118864482230380891724525022,
0.13118864482230380891724525022, 1.82410748104880055423200393360, 2.19599488161042912285673705934, 3.56959499915042015431233962641, 4.27440674080925693175285995435, 4.98145696792792449552037861277, 6.04781955065939962990260466204, 6.80932447300415709737439928025, 7.20632461540460542280761628684, 8.115219961734283907776990007599
