| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 16·5-s + 6·6-s + 28·7-s − 8·8-s + 9·9-s + 32·10-s + 34·11-s − 12·12-s − 13·13-s − 56·14-s + 48·15-s + 16·16-s + 138·17-s − 18·18-s + 108·19-s − 64·20-s − 84·21-s − 68·22-s − 52·23-s + 24·24-s + 131·25-s + 26·26-s − 27·27-s + 112·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.43·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.01·10-s + 0.931·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 0.826·15-s + 1/4·16-s + 1.96·17-s − 0.235·18-s + 1.30·19-s − 0.715·20-s − 0.872·21-s − 0.658·22-s − 0.471·23-s + 0.204·24-s + 1.04·25-s + 0.196·26-s − 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8688481197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8688481197\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 16 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 138 T + p^{3} T^{2} \) |
| 19 | \( 1 - 108 T + p^{3} T^{2} \) |
| 23 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 + 176 T + p^{3} T^{2} \) |
| 37 | \( 1 - 342 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 140 T + p^{3} T^{2} \) |
| 47 | \( 1 - 454 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 + 154 T + p^{3} T^{2} \) |
| 61 | \( 1 - 34 T + p^{3} T^{2} \) |
| 67 | \( 1 + 656 T + p^{3} T^{2} \) |
| 71 | \( 1 - 550 T + p^{3} T^{2} \) |
| 73 | \( 1 - 614 T + p^{3} T^{2} \) |
| 79 | \( 1 - 8 T + p^{3} T^{2} \) |
| 83 | \( 1 - 762 T + p^{3} T^{2} \) |
| 89 | \( 1 + 444 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1022 T + p^{3} T^{2} \) |
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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
−14.36187652030374236119641714348, −12.16390880733516895682194854980, −11.70689693897565924586307523323, −10.91830244627118822725205067042, −9.419328241569117798848743953908, −7.85370014128061745764062675793, −7.49990699700038422974083219008, −5.46429382967118041432651062610, −3.89437210774905401661320507711, −1.10400107719111997753143848713,
1.10400107719111997753143848713, 3.89437210774905401661320507711, 5.46429382967118041432651062610, 7.49990699700038422974083219008, 7.85370014128061745764062675793, 9.419328241569117798848743953908, 10.91830244627118822725205067042, 11.70689693897565924586307523323, 12.16390880733516895682194854980, 14.36187652030374236119641714348
