| L(s) = 1 | − 1.73·2-s + 0.999·4-s − 1.73·5-s + 1.73·8-s + 2.99·10-s + 3.46·11-s + 13-s − 5·16-s + 5.19·17-s − 2·19-s − 1.73·20-s − 5.99·22-s − 3.46·23-s − 2.00·25-s − 1.73·26-s − 1.73·29-s − 8·31-s + 5.19·32-s − 9·34-s − 7·37-s + 3.46·38-s − 3.00·40-s − 6.92·41-s + 2·43-s + 3.46·44-s + 5.99·46-s + 6.92·47-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s − 0.774·5-s + 0.612·8-s + 0.948·10-s + 1.04·11-s + 0.277·13-s − 1.25·16-s + 1.26·17-s − 0.458·19-s − 0.387·20-s − 1.27·22-s − 0.722·23-s − 0.400·25-s − 0.339·26-s − 0.321·29-s − 1.43·31-s + 0.918·32-s − 1.54·34-s − 1.15·37-s + 0.561·38-s − 0.474·40-s − 1.08·41-s + 0.304·43-s + 0.522·44-s + 0.884·46-s + 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.351864881282740620795146992606, −7.41567786564871440930638373025, −7.07155350805120759726058089148, −6.01899900172669675147831817346, −5.12939503069937808715385834084, −3.92992721624233989312251724769, −3.67277783861965739710948188494, −2.07487228904507212952730528348, −1.17194717343387249432052540940, 0,
1.17194717343387249432052540940, 2.07487228904507212952730528348, 3.67277783861965739710948188494, 3.92992721624233989312251724769, 5.12939503069937808715385834084, 6.01899900172669675147831817346, 7.07155350805120759726058089148, 7.41567786564871440930638373025, 8.351864881282740620795146992606
