L(s) = 1 | + 2-s − 1.87·3-s + 4-s + 2·5-s − 1.87·6-s + 5.06·7-s + 8-s + 0.532·9-s + 2·10-s − 1.41·11-s − 1.87·12-s − 1.30·13-s + 5.06·14-s − 3.75·15-s + 16-s + 2.38·17-s + 0.532·18-s + 2·20-s − 9.51·21-s − 1.41·22-s − 3.06·23-s − 1.87·24-s − 25-s − 1.30·26-s + 4.63·27-s + 5.06·28-s + 8.45·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.08·3-s + 0.5·4-s + 0.894·5-s − 0.767·6-s + 1.91·7-s + 0.353·8-s + 0.177·9-s + 0.632·10-s − 0.425·11-s − 0.542·12-s − 0.362·13-s + 1.35·14-s − 0.970·15-s + 0.250·16-s + 0.579·17-s + 0.125·18-s + 0.447·20-s − 2.07·21-s − 0.300·22-s − 0.638·23-s − 0.383·24-s − 0.200·25-s − 0.256·26-s + 0.892·27-s + 0.957·28-s + 1.56·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.219448151\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.219448151\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 - 8.45T + 29T^{2} \) |
| 31 | \( 1 - 0.369T + 31T^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 + 1.53T + 41T^{2} \) |
| 43 | \( 1 + 0.758T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 1.67T + 53T^{2} \) |
| 59 | \( 1 - 0.716T + 59T^{2} \) |
| 61 | \( 1 - 9.75T + 61T^{2} \) |
| 67 | \( 1 + 1.40T + 67T^{2} \) |
| 71 | \( 1 + 6.36T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 + 3.98T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.53T + 97T^{2} \) |
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
−10.54800782691874618405543867314, −9.989565544662522711752690013270, −8.473229390319631569558225690595, −7.74027486224275091101613003797, −6.55317285559256045478504321872, −5.64050138065177359492559229806, −5.14834280382561927379914243387, −4.39646983391725764260083786340, −2.55918184895952397256267162215, −1.39036145962278626257464958440,
1.39036145962278626257464958440, 2.55918184895952397256267162215, 4.39646983391725764260083786340, 5.14834280382561927379914243387, 5.64050138065177359492559229806, 6.55317285559256045478504321872, 7.74027486224275091101613003797, 8.473229390319631569558225690595, 9.989565544662522711752690013270, 10.54800782691874618405543867314