| L(s) = 1 | + 5.09·2-s − 3.26·3-s + 18.0·4-s − 18.2·5-s − 16.6·6-s − 29.9·7-s + 51.0·8-s − 16.3·9-s − 92.9·10-s − 11·11-s − 58.7·12-s − 152.·14-s + 59.5·15-s + 116.·16-s + 19.4·17-s − 83.3·18-s − 94.9·19-s − 328.·20-s + 97.8·21-s − 56.0·22-s − 179.·23-s − 166.·24-s + 207.·25-s + 141.·27-s − 539.·28-s + 90.3·29-s + 303.·30-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 0.628·3-s + 2.25·4-s − 1.63·5-s − 1.13·6-s − 1.61·7-s + 2.25·8-s − 0.605·9-s − 2.94·10-s − 0.301·11-s − 1.41·12-s − 2.91·14-s + 1.02·15-s + 1.81·16-s + 0.277·17-s − 1.09·18-s − 1.14·19-s − 3.67·20-s + 1.01·21-s − 0.543·22-s − 1.62·23-s − 1.41·24-s + 1.66·25-s + 1.00·27-s − 3.64·28-s + 0.578·29-s + 1.84·30-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
| \(L(2)\) |
\(\approx\) |
\(1.470737332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470737332\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.09T + 8T^{2} \) |
| 3 | \( 1 + 3.26T + 27T^{2} \) |
| 5 | \( 1 + 18.2T + 125T^{2} \) |
| 7 | \( 1 + 29.9T + 343T^{2} \) |
| 17 | \( 1 - 19.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 179.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 90.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 259.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 305.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 222.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 36.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 175.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 173.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 773.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 816.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 88.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 276.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 376.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 793.T + 9.12e5T^{2} \) |
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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
−8.621034826772561380070124304007, −7.80650312333895671182873997945, −6.84884663641856372383417406225, −6.29785930403631380214689726150, −5.70427726540346363602860075305, −4.57871349701227044071811376007, −4.00906645383346447400441873166, −3.24046370286137943725279240036, −2.56103587971777888647895225344, −0.41057833914511877355750151860,
0.41057833914511877355750151860, 2.56103587971777888647895225344, 3.24046370286137943725279240036, 4.00906645383346447400441873166, 4.57871349701227044071811376007, 5.70427726540346363602860075305, 6.29785930403631380214689726150, 6.84884663641856372383417406225, 7.80650312333895671182873997945, 8.621034826772561380070124304007
