| L(s) = 1 | + 1.50·3-s − 3.76·5-s + 4.72·7-s − 0.734·9-s − 0.374·13-s − 5.66·15-s − 2.41·17-s + 19-s + 7.10·21-s − 4.04·23-s + 9.17·25-s − 5.62·27-s + 8.79·29-s − 5.56·31-s − 17.7·35-s − 4.35·37-s − 0.563·39-s + 1.30·41-s + 7.35·43-s + 2.76·45-s − 1.25·47-s + 15.3·49-s − 3.63·51-s − 12.6·53-s + 1.50·57-s − 0.988·59-s + 14.7·61-s + ⋯ |
| L(s) = 1 | + 0.869·3-s − 1.68·5-s + 1.78·7-s − 0.244·9-s − 0.103·13-s − 1.46·15-s − 0.586·17-s + 0.229·19-s + 1.55·21-s − 0.842·23-s + 1.83·25-s − 1.08·27-s + 1.63·29-s − 0.999·31-s − 3.00·35-s − 0.715·37-s − 0.0902·39-s + 0.204·41-s + 1.12·43-s + 0.412·45-s − 0.182·47-s + 2.18·49-s − 0.509·51-s − 1.74·53-s + 0.199·57-s − 0.128·59-s + 1.89·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.50T + 3T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 7 | \( 1 - 4.72T + 7T^{2} \) |
| 13 | \( 1 + 0.374T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 - 8.79T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 + 4.35T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 - 7.35T + 43T^{2} \) |
| 47 | \( 1 + 1.25T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 0.988T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 1.84T + 71T^{2} \) |
| 73 | \( 1 + 8.51T + 73T^{2} \) |
| 79 | \( 1 + 2.81T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 7.66T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
| 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
−7.58015992391874585574824376878, −7.13355957057024570855997784785, −5.97164029539227885985538482653, −5.03181490690687398563743679043, −4.43823311623701049176320751524, −3.92707754069971558490027982978, −3.08689346663036440567430833604, −2.27834113107156348127454489649, −1.31394046144142965622608683627, 0,
1.31394046144142965622608683627, 2.27834113107156348127454489649, 3.08689346663036440567430833604, 3.92707754069971558490027982978, 4.43823311623701049176320751524, 5.03181490690687398563743679043, 5.97164029539227885985538482653, 7.13355957057024570855997784785, 7.58015992391874585574824376878
