| L(s) = 1 | − 5.61·2-s + 3·3-s + 23.4·4-s − 5·5-s − 16.8·6-s − 29.7·7-s − 86.8·8-s + 9·9-s + 28.0·10-s + 70.4·12-s + 44.6·13-s + 166.·14-s − 15·15-s + 299.·16-s + 30.3·17-s − 50.4·18-s − 64.6·19-s − 117.·20-s − 89.2·21-s − 69.8·23-s − 260.·24-s + 25·25-s − 250.·26-s + 27·27-s − 698.·28-s − 34.5·29-s + 84.1·30-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 0.577·3-s + 2.93·4-s − 0.447·5-s − 1.14·6-s − 1.60·7-s − 3.83·8-s + 0.333·9-s + 0.887·10-s + 1.69·12-s + 0.953·13-s + 3.18·14-s − 0.258·15-s + 4.67·16-s + 0.433·17-s − 0.661·18-s − 0.780·19-s − 1.31·20-s − 0.927·21-s − 0.633·23-s − 2.21·24-s + 0.200·25-s − 1.89·26-s + 0.192·27-s − 4.71·28-s − 0.221·29-s + 0.512·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 5.61T + 8T^{2} \) |
| 7 | \( 1 + 29.7T + 343T^{2} \) |
| 13 | \( 1 - 44.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 30.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 69.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 34.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 268.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 449.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 0.808T + 1.03e5T^{2} \) |
| 53 | \( 1 - 128.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 191.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 531.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 368.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 252.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 537.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 396.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 206.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−8.675434001287497698487028638945, −7.966223043762112420715388344643, −7.17314422798231864588746781466, −6.48270854151970532830892325721, −5.90062743662270292194328156513, −3.79132030051792198676978021137, −3.11989335130383453094216941847, −2.20097175095767107100233554151, −0.967074561951979324278584303532, 0,
0.967074561951979324278584303532, 2.20097175095767107100233554151, 3.11989335130383453094216941847, 3.79132030051792198676978021137, 5.90062743662270292194328156513, 6.48270854151970532830892325721, 7.17314422798231864588746781466, 7.966223043762112420715388344643, 8.675434001287497698487028638945
