| L(s) = 1 | + 2.63·2-s + 3-s + 4.94·4-s − 1.82·5-s + 2.63·6-s − 7-s + 7.77·8-s + 9-s − 4.81·10-s − 2.90·11-s + 4.94·12-s + 0.557·13-s − 2.63·14-s − 1.82·15-s + 10.5·16-s + 1.40·17-s + 2.63·18-s + 3.95·19-s − 9.03·20-s − 21-s − 7.65·22-s + 0.0518·23-s + 7.77·24-s − 1.66·25-s + 1.46·26-s + 27-s − 4.94·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 0.577·3-s + 2.47·4-s − 0.816·5-s + 1.07·6-s − 0.377·7-s + 2.74·8-s + 0.333·9-s − 1.52·10-s − 0.875·11-s + 1.42·12-s + 0.154·13-s − 0.704·14-s − 0.471·15-s + 2.64·16-s + 0.339·17-s + 0.621·18-s + 0.907·19-s − 2.02·20-s − 0.218·21-s − 1.63·22-s + 0.0108·23-s + 1.58·24-s − 0.333·25-s + 0.288·26-s + 0.192·27-s − 0.935·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.738603159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.738603159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 - 0.557T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 - 0.0518T + 23T^{2} \) |
| 29 | \( 1 - 8.07T + 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 - 6.44T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 - 4.86T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 0.312T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + 4.96T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−7.65423073624745681160767885801, −7.09920664749129243417730518118, −6.21182618396177224266178052080, −5.69646175508076327964479427457, −4.76977780946279290452303317935, −4.30976512431717020508016681333, −3.55835039567948845052375885819, −2.88473053044538003379999981043, −2.45136133819134642250937823165, −1.03469389317498035865223599255,
1.03469389317498035865223599255, 2.45136133819134642250937823165, 2.88473053044538003379999981043, 3.55835039567948845052375885819, 4.30976512431717020508016681333, 4.76977780946279290452303317935, 5.69646175508076327964479427457, 6.21182618396177224266178052080, 7.09920664749129243417730518118, 7.65423073624745681160767885801
