| L(s) = 1 | − 10.5·5-s + 22·7-s + 5.29·11-s + 13·13-s + 116.·17-s + 126·19-s − 31.7·23-s − 12.9·25-s + 52.9·29-s + 182·31-s − 232.·35-s − 86·37-s − 444.·41-s − 96·43-s − 365.·47-s + 141·49-s − 190.·53-s − 56.0·55-s + 587.·59-s + 574·61-s − 137.·65-s + 530·67-s − 809.·71-s − 154·73-s + 116.·77-s + 460·79-s + 322.·83-s + ⋯ |
| L(s) = 1 | − 0.946·5-s + 1.18·7-s + 0.145·11-s + 0.277·13-s + 1.66·17-s + 1.52·19-s − 0.287·23-s − 0.103·25-s + 0.338·29-s + 1.05·31-s − 1.12·35-s − 0.382·37-s − 1.69·41-s − 0.340·43-s − 1.13·47-s + 0.411·49-s − 0.493·53-s − 0.137·55-s + 1.29·59-s + 1.20·61-s − 0.262·65-s + 0.966·67-s − 1.35·71-s − 0.246·73-s + 0.172·77-s + 0.655·79-s + 0.426·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.423744734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.423744734\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 10.5T + 125T^{2} \) |
| 7 | \( 1 - 22T + 343T^{2} \) |
| 11 | \( 1 - 5.29T + 1.33e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 126T + 6.85e3T^{2} \) |
| 23 | \( 1 + 31.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 182T + 2.97e4T^{2} \) |
| 37 | \( 1 + 86T + 5.06e4T^{2} \) |
| 41 | \( 1 + 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 96T + 7.95e4T^{2} \) |
| 47 | \( 1 + 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 587.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 574T + 2.26e5T^{2} \) |
| 67 | \( 1 - 530T + 3.00e5T^{2} \) |
| 71 | \( 1 + 809.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154T + 3.89e5T^{2} \) |
| 79 | \( 1 - 460T + 4.93e5T^{2} \) |
| 83 | \( 1 - 322.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 70T + 9.12e5T^{2} \) |
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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
−8.587312073348037404643404087014, −7.998373767938106964748992131891, −7.59398403599394666927823793813, −6.60474381323440160411632242825, −5.40781936592282316597809230681, −4.90475345646598451914629993566, −3.80500945247167310784569168428, −3.14702860015315655433302913679, −1.65302357910904430810153014684, −0.78135240449665849959619226711,
0.78135240449665849959619226711, 1.65302357910904430810153014684, 3.14702860015315655433302913679, 3.80500945247167310784569168428, 4.90475345646598451914629993566, 5.40781936592282316597809230681, 6.60474381323440160411632242825, 7.59398403599394666927823793813, 7.998373767938106964748992131891, 8.587312073348037404643404087014
