| L(s) = 1 | − 0.138·3-s − 10.0·5-s − 26.9·9-s + 18.6·11-s + 10.0·13-s + 1.39·15-s + 73.6·17-s − 76.0·19-s − 146.·23-s − 23.2·25-s + 7.46·27-s − 157.·29-s − 69.8·31-s − 2.58·33-s + 309.·37-s − 1.39·39-s + 482.·41-s − 17.3·43-s + 272.·45-s + 346.·47-s − 10.1·51-s − 73.4·53-s − 188.·55-s + 10.5·57-s − 704.·59-s + 841.·61-s − 101.·65-s + ⋯ |
| L(s) = 1 | − 0.0265·3-s − 0.902·5-s − 0.999·9-s + 0.511·11-s + 0.215·13-s + 0.0239·15-s + 1.05·17-s − 0.918·19-s − 1.33·23-s − 0.186·25-s + 0.0531·27-s − 1.00·29-s − 0.404·31-s − 0.0136·33-s + 1.37·37-s − 0.00572·39-s + 1.83·41-s − 0.0615·43-s + 0.901·45-s + 1.07·47-s − 0.0279·51-s − 0.190·53-s − 0.461·55-s + 0.0244·57-s − 1.55·59-s + 1.76·61-s − 0.194·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.221995487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221995487\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.138T + 27T^{2} \) |
| 5 | \( 1 + 10.0T + 125T^{2} \) |
| 11 | \( 1 - 18.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 146.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 69.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 482.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 17.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 73.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 704.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 841.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 891.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 525.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 376.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 771.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 175.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.844797162963070128838311981322, −8.987778183771716825536294901211, −8.066197439068803961971006857213, −7.58018645773102097401355317691, −6.25981022387122667658971860763, −5.61411064214104715311682231146, −4.21829720466458733691387742447, −3.56806707885836838314432520945, −2.25082312441583864804486466970, −0.60567352592223929778347721083,
0.60567352592223929778347721083, 2.25082312441583864804486466970, 3.56806707885836838314432520945, 4.21829720466458733691387742447, 5.61411064214104715311682231146, 6.25981022387122667658971860763, 7.58018645773102097401355317691, 8.066197439068803961971006857213, 8.987778183771716825536294901211, 9.844797162963070128838311981322
