| L(s) = 1 | − 2.02·2-s − 9.13·3-s − 3.88·4-s − 2.96·5-s + 18.5·6-s − 15.4·7-s + 24.1·8-s + 56.3·9-s + 6.02·10-s − 11·11-s + 35.4·12-s + 31.2·14-s + 27.1·15-s − 17.8·16-s − 83.1·17-s − 114.·18-s + 14.0·19-s + 11.5·20-s + 140.·21-s + 22.3·22-s − 78.2·23-s − 220.·24-s − 116.·25-s − 268.·27-s + 59.9·28-s − 4.38·29-s − 54.9·30-s + ⋯ |
| L(s) = 1 | − 0.717·2-s − 1.75·3-s − 0.485·4-s − 0.265·5-s + 1.26·6-s − 0.832·7-s + 1.06·8-s + 2.08·9-s + 0.190·10-s − 0.301·11-s + 0.853·12-s + 0.597·14-s + 0.466·15-s − 0.278·16-s − 1.18·17-s − 1.49·18-s + 0.169·19-s + 0.128·20-s + 1.46·21-s + 0.216·22-s − 0.709·23-s − 1.87·24-s − 0.929·25-s − 1.91·27-s + 0.404·28-s − 0.0280·29-s − 0.334·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.02T + 8T^{2} \) |
| 3 | \( 1 + 9.13T + 27T^{2} \) |
| 5 | \( 1 + 2.96T + 125T^{2} \) |
| 7 | \( 1 + 15.4T + 343T^{2} \) |
| 17 | \( 1 + 83.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 4.38T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 391.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 91.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 500.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 585.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 207.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 211.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 408.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 221.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 692.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 603.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 979.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.61e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−8.624055776314642660244323317128, −7.53163561104263824321243074303, −6.94018301984670837981405382050, −6.05656824007401333884785364490, −5.36621986998425812088315052354, −4.47476936734560128835361908202, −3.76299610078055874016230504191, −1.95692316299869806170232743688, −0.63984231792750604642910804703, 0,
0.63984231792750604642910804703, 1.95692316299869806170232743688, 3.76299610078055874016230504191, 4.47476936734560128835361908202, 5.36621986998425812088315052354, 6.05656824007401333884785364490, 6.94018301984670837981405382050, 7.53163561104263824321243074303, 8.624055776314642660244323317128
