L(s) = 1 | − 2.24·2-s + 3.03·4-s + 5-s + 1.09·7-s − 2.31·8-s − 2.24·10-s + 3.76·11-s − 6.02·13-s − 2.46·14-s − 0.869·16-s − 3.14·17-s + 0.176·19-s + 3.03·20-s − 8.45·22-s + 0.808·23-s + 25-s + 13.5·26-s + 3.33·28-s + 5.42·29-s + 3.31·31-s + 6.58·32-s + 7.05·34-s + 1.09·35-s − 11.5·37-s − 0.396·38-s − 2.31·40-s + 1.30·41-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 1.51·4-s + 0.447·5-s + 0.415·7-s − 0.818·8-s − 0.709·10-s + 1.13·11-s − 1.66·13-s − 0.658·14-s − 0.217·16-s − 0.762·17-s + 0.0405·19-s + 0.678·20-s − 1.80·22-s + 0.168·23-s + 0.200·25-s + 2.64·26-s + 0.629·28-s + 1.00·29-s + 0.596·31-s + 1.16·32-s + 1.21·34-s + 0.185·35-s − 1.89·37-s − 0.0643·38-s − 0.366·40-s + 0.203·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 11 | \( 1 - 3.76T + 11T^{2} \) |
| 13 | \( 1 + 6.02T + 13T^{2} \) |
| 17 | \( 1 + 3.14T + 17T^{2} \) |
| 19 | \( 1 - 0.176T + 19T^{2} \) |
| 23 | \( 1 - 0.808T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 - 3.31T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 0.0238T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + 7.11T + 53T^{2} \) |
| 59 | \( 1 + 9.73T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 - 3.36T + 71T^{2} \) |
| 73 | \( 1 - 6.82T + 73T^{2} \) |
| 79 | \( 1 - 6.52T + 79T^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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
−8.222709363981938233751846754941, −7.48915694547118141034218280020, −6.80932443859109450505083829528, −6.31229630420273991990681051960, −5.01316841405774376335772106538, −4.44321735584031583055367662018, −2.99206078025851613994106614627, −2.06277500183195424395542568159, −1.32797050202936110129523365619, 0,
1.32797050202936110129523365619, 2.06277500183195424395542568159, 2.99206078025851613994106614627, 4.44321735584031583055367662018, 5.01316841405774376335772106538, 6.31229630420273991990681051960, 6.80932443859109450505083829528, 7.48915694547118141034218280020, 8.222709363981938233751846754941