| L(s) = 1 | − 4.45·2-s + 11.8·4-s + 12.1·5-s + 7·7-s − 17.2·8-s − 54.3·10-s + 11·11-s − 15.6·13-s − 31.1·14-s − 18.1·16-s + 43.3·17-s + 51.7·19-s + 144.·20-s − 49.0·22-s + 121.·23-s + 23.6·25-s + 69.7·26-s + 83.0·28-s + 187.·29-s + 91.9·31-s + 218.·32-s − 193.·34-s + 85.3·35-s − 226.·37-s − 230.·38-s − 209.·40-s − 11.2·41-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 1.48·4-s + 1.09·5-s + 0.377·7-s − 0.760·8-s − 1.71·10-s + 0.301·11-s − 0.333·13-s − 0.595·14-s − 0.283·16-s + 0.618·17-s + 0.625·19-s + 1.61·20-s − 0.475·22-s + 1.09·23-s + 0.188·25-s + 0.526·26-s + 0.560·28-s + 1.19·29-s + 0.532·31-s + 1.20·32-s − 0.973·34-s + 0.412·35-s − 1.00·37-s − 0.985·38-s − 0.829·40-s − 0.0430·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.291178189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291178189\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 4.45T + 8T^{2} \) |
| 5 | \( 1 - 12.1T + 125T^{2} \) |
| 13 | \( 1 + 15.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 51.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 91.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 226.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 11.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 98.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 487.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 697.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 486.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 436.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 715.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 860.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 397.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 237.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 590.T + 9.12e5T^{2} \) |
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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
−10.07428954119634228263837995444, −9.173638662485378044106422717431, −8.614490607159946114126066581415, −7.58125019211478982373393901256, −6.84555090430115715223854636795, −5.80969932318821890562048343018, −4.74430683349647594119378387895, −2.92603031935082200278204693207, −1.77843542542002214978940912404, −0.884472823845140951219432637219,
0.884472823845140951219432637219, 1.77843542542002214978940912404, 2.92603031935082200278204693207, 4.74430683349647594119378387895, 5.80969932318821890562048343018, 6.84555090430115715223854636795, 7.58125019211478982373393901256, 8.614490607159946114126066581415, 9.173638662485378044106422717431, 10.07428954119634228263837995444
