| L(s) = 1 | + 1.82·2-s − 3·3-s − 4.65·4-s − 7.31·5-s − 5.48·6-s − 11.1·7-s − 23.1·8-s + 9·9-s − 13.3·10-s − 16.2·11-s + 13.9·12-s − 13.0·13-s − 20.4·14-s + 21.9·15-s − 5.05·16-s + 33.8·17-s + 16.4·18-s − 6.11·19-s + 34.0·20-s + 33.5·21-s − 29.6·22-s + 23·23-s + 69.4·24-s − 71.5·25-s − 23.8·26-s − 27·27-s + 52.0·28-s + ⋯ |
| L(s) = 1 | + 0.646·2-s − 0.577·3-s − 0.582·4-s − 0.654·5-s − 0.373·6-s − 0.603·7-s − 1.02·8-s + 0.333·9-s − 0.422·10-s − 0.444·11-s + 0.336·12-s − 0.277·13-s − 0.389·14-s + 0.377·15-s − 0.0790·16-s + 0.483·17-s + 0.215·18-s − 0.0738·19-s + 0.380·20-s + 0.348·21-s − 0.287·22-s + 0.208·23-s + 0.590·24-s − 0.572·25-s − 0.179·26-s − 0.192·27-s + 0.351·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 1.82T + 8T^{2} \) |
| 5 | \( 1 + 7.31T + 125T^{2} \) |
| 7 | \( 1 + 11.1T + 343T^{2} \) |
| 11 | \( 1 + 16.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.11T + 6.85e3T^{2} \) |
| 29 | \( 1 - 84.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 63.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 75.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 260.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 224.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 44.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 466.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 520.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 906.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 920.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 143.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−13.52707007039510107015081096236, −12.59982429997064407357698436529, −11.80888687744404231142806594387, −10.38373694528959717892476606986, −9.140319020221247658203111187603, −7.62049942676746409279355280780, −6.05431800823528768561480565422, −4.81080656172370216045629619196, −3.44469640446503578637648599418, 0,
3.44469640446503578637648599418, 4.81080656172370216045629619196, 6.05431800823528768561480565422, 7.62049942676746409279355280780, 9.140319020221247658203111187603, 10.38373694528959717892476606986, 11.80888687744404231142806594387, 12.59982429997064407357698436529, 13.52707007039510107015081096236
