| L(s) = 1 | + 5.80·5-s + 26.1·7-s − 37.2·11-s − 30.3·13-s + 48.4·17-s − 88.1·19-s + 84.5·23-s − 91.3·25-s − 176.·29-s − 155.·31-s + 151.·35-s − 258.·37-s − 220.·41-s + 47.7·43-s + 129.·47-s + 339.·49-s − 577.·53-s − 216.·55-s − 243.·59-s − 687.·61-s − 176.·65-s − 378.·67-s + 332.·71-s + 1.07e3·73-s − 972.·77-s + 1.26e3·79-s + 1.22e3·83-s + ⋯ |
| L(s) = 1 | + 0.519·5-s + 1.41·7-s − 1.02·11-s − 0.647·13-s + 0.691·17-s − 1.06·19-s + 0.766·23-s − 0.730·25-s − 1.13·29-s − 0.900·31-s + 0.732·35-s − 1.14·37-s − 0.839·41-s + 0.169·43-s + 0.401·47-s + 0.989·49-s − 1.49·53-s − 0.529·55-s − 0.537·59-s − 1.44·61-s − 0.335·65-s − 0.689·67-s + 0.556·71-s + 1.71·73-s − 1.43·77-s + 1.79·79-s + 1.62·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 5.80T + 125T^{2} \) |
| 7 | \( 1 - 26.1T + 343T^{2} \) |
| 11 | \( 1 + 37.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 84.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 176.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 220.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 47.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 577.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 243.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 687.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 378.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 332.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−8.881112567412567536357205181071, −7.88078799674396141632786638749, −7.54852897053740977349269227663, −6.31222767267013678073065789878, −5.22192927302015279859251481490, −4.95397879350713460949522860977, −3.62443649769964574448618125958, −2.30024999448435967644104714555, −1.61214374578738884529178433652, 0,
1.61214374578738884529178433652, 2.30024999448435967644104714555, 3.62443649769964574448618125958, 4.95397879350713460949522860977, 5.22192927302015279859251481490, 6.31222767267013678073065789878, 7.54852897053740977349269227663, 7.88078799674396141632786638749, 8.881112567412567536357205181071
