L(s) = 1 | + 2-s − 2.55·3-s + 4-s − 3.59·5-s − 2.55·6-s − 3.65·7-s + 8-s + 3.51·9-s − 3.59·10-s + 11-s − 2.55·12-s − 1.76·13-s − 3.65·14-s + 9.17·15-s + 16-s + 5.45·17-s + 3.51·18-s − 3.47·19-s − 3.59·20-s + 9.31·21-s + 22-s + 1.38·23-s − 2.55·24-s + 7.93·25-s − 1.76·26-s − 1.30·27-s − 3.65·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.47·3-s + 0.5·4-s − 1.60·5-s − 1.04·6-s − 1.37·7-s + 0.353·8-s + 1.17·9-s − 1.13·10-s + 0.301·11-s − 0.736·12-s − 0.488·13-s − 0.975·14-s + 2.36·15-s + 0.250·16-s + 1.32·17-s + 0.827·18-s − 0.798·19-s − 0.804·20-s + 2.03·21-s + 0.213·22-s + 0.288·23-s − 0.520·24-s + 1.58·25-s − 0.345·26-s − 0.251·27-s − 0.689·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
good | 3 | \( 1 + 2.55T + 3T^{2} \) |
| 5 | \( 1 + 3.59T + 5T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 - 4.84T + 29T^{2} \) |
| 31 | \( 1 - 5.30T + 31T^{2} \) |
| 37 | \( 1 - 7.29T + 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 3.27T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 - 1.97T + 67T^{2} \) |
| 71 | \( 1 - 0.420T + 71T^{2} \) |
| 73 | \( 1 + 4.86T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 8.57T + 97T^{2} \) |
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
−7.65792795720410446813117037547, −7.12171739396200532568645970269, −6.27599894289253769794318938346, −6.03510879649128211975831692850, −4.82703247317078492249165827367, −4.44584402566541541952773983411, −3.49043837541219112353908545980, −2.89598268254502794547486662214, −0.965791379265017190297019750906, 0,
0.965791379265017190297019750906, 2.89598268254502794547486662214, 3.49043837541219112353908545980, 4.44584402566541541952773983411, 4.82703247317078492249165827367, 6.03510879649128211975831692850, 6.27599894289253769794318938346, 7.12171739396200532568645970269, 7.65792795720410446813117037547