| L(s) = 1 | + 2-s − 3-s + 4-s + 2.25·5-s − 6-s + 3.43·7-s + 8-s + 9-s + 2.25·10-s + 4.91·11-s − 12-s + 2.89·13-s + 3.43·14-s − 2.25·15-s + 16-s − 17-s + 18-s + 1.28·19-s + 2.25·20-s − 3.43·21-s + 4.91·22-s + 1.07·23-s − 24-s + 0.0704·25-s + 2.89·26-s − 27-s + 3.43·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.00·5-s − 0.408·6-s + 1.29·7-s + 0.353·8-s + 0.333·9-s + 0.712·10-s + 1.48·11-s − 0.288·12-s + 0.802·13-s + 0.918·14-s − 0.581·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.295·19-s + 0.503·20-s − 0.749·21-s + 1.04·22-s + 0.224·23-s − 0.204·24-s + 0.0140·25-s + 0.567·26-s − 0.192·27-s + 0.649·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.648189490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.648189490\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 2.25T + 5T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 + 8.07T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 4.17T + 43T^{2} \) |
| 47 | \( 1 - 3.20T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 61 | \( 1 + 6.05T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 + 5.30T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−7.983994445989674974614942567027, −7.16258046051293120693006202608, −6.35311022676010899076890634364, −5.95095949005323329509761146769, −5.28013505327259272284395179596, −4.44054683783483415281008451123, −3.97654548739937435308123897944, −2.75070915590016032023586819177, −1.61755516512236721951446419328, −1.28009256159153936207391914742,
1.28009256159153936207391914742, 1.61755516512236721951446419328, 2.75070915590016032023586819177, 3.97654548739937435308123897944, 4.44054683783483415281008451123, 5.28013505327259272284395179596, 5.95095949005323329509761146769, 6.35311022676010899076890634364, 7.16258046051293120693006202608, 7.983994445989674974614942567027
