| L(s) = 1 | + 4.77·2-s − 3·3-s + 14.8·4-s + 17.5·5-s − 14.3·6-s − 4.31·7-s + 32.4·8-s + 9·9-s + 84.0·10-s + 63.1·11-s − 44.4·12-s − 49.7·13-s − 20.6·14-s − 52.7·15-s + 36.7·16-s + 26.8·17-s + 42.9·18-s + 260.·20-s + 12.9·21-s + 301.·22-s + 96.9·23-s − 97.4·24-s + 184.·25-s − 237.·26-s − 27·27-s − 63.9·28-s + 235.·29-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.85·4-s + 1.57·5-s − 0.974·6-s − 0.233·7-s + 1.43·8-s + 0.333·9-s + 2.65·10-s + 1.73·11-s − 1.06·12-s − 1.06·13-s − 0.393·14-s − 0.908·15-s + 0.574·16-s + 0.382·17-s + 0.562·18-s + 2.91·20-s + 0.134·21-s + 2.92·22-s + 0.879·23-s − 0.829·24-s + 1.47·25-s − 1.79·26-s − 0.192·27-s − 0.431·28-s + 1.50·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.172225051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.172225051\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 4.77T + 8T^{2} \) |
| 5 | \( 1 - 17.5T + 125T^{2} \) |
| 7 | \( 1 + 4.31T + 343T^{2} \) |
| 11 | \( 1 - 63.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 96.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 103.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 89.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 398.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 14.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 420.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 526.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 561.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 231.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 78.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 40.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 44.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 761.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 150.T + 9.12e5T^{2} \) |
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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
−9.685514270667356475879866170689, −8.933227865991205758456071085906, −7.13566889579327134736518728704, −6.56144815670779867113433239477, −5.98793034781194395355191073850, −5.18018097813713369064523848884, −4.51207703827931614318396204732, −3.34360942201699751591156134437, −2.30299362548980063433114285548, −1.25835874240815397589422699113,
1.25835874240815397589422699113, 2.30299362548980063433114285548, 3.34360942201699751591156134437, 4.51207703827931614318396204732, 5.18018097813713369064523848884, 5.98793034781194395355191073850, 6.56144815670779867113433239477, 7.13566889579327134736518728704, 8.933227865991205758456071085906, 9.685514270667356475879866170689
