L(s) = 1 | + 0.539·3-s − 3.80·7-s − 2.70·9-s + 2.34·11-s + 4.04·13-s − 6.78·17-s + 5.46·19-s − 2.04·21-s + 4.34·23-s − 3.07·27-s + 7.75·29-s − 1.61·31-s + 1.26·33-s + 37-s + 2.18·39-s − 3.07·41-s − 11.7·43-s + 2.14·47-s + 7.44·49-s − 3.65·51-s − 14.0·53-s + 2.94·57-s − 7.37·59-s + 14.9·61-s + 10.2·63-s − 14.1·67-s + 2.34·69-s + ⋯ |
L(s) = 1 | + 0.311·3-s − 1.43·7-s − 0.903·9-s + 0.705·11-s + 1.12·13-s − 1.64·17-s + 1.25·19-s − 0.447·21-s + 0.904·23-s − 0.592·27-s + 1.44·29-s − 0.290·31-s + 0.219·33-s + 0.164·37-s + 0.349·39-s − 0.480·41-s − 1.79·43-s + 0.312·47-s + 1.06·49-s − 0.512·51-s − 1.93·53-s + 0.389·57-s − 0.960·59-s + 1.91·61-s + 1.29·63-s − 1.72·67-s + 0.281·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 0.539T + 3T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 - 7.75T + 29T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 41 | \( 1 + 3.07T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 2.14T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 + 7.37T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 4.34T + 73T^{2} \) |
| 79 | \( 1 + 4.72T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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
−8.494827828574159972519117922880, −7.28441138019843498069662611685, −6.49379309327822519666823008377, −6.22565563770032880328813992957, −5.17560766099022741925379977687, −4.15014197740312853741741122781, −3.21039023559724378301683672903, −2.87854488724874381190793522315, −1.40633438816869607396352803674, 0,
1.40633438816869607396352803674, 2.87854488724874381190793522315, 3.21039023559724378301683672903, 4.15014197740312853741741122781, 5.17560766099022741925379977687, 6.22565563770032880328813992957, 6.49379309327822519666823008377, 7.28441138019843498069662611685, 8.494827828574159972519117922880