L(s) = 1 | − 1.39·2-s − 3-s − 0.0671·4-s + 1.39·6-s + 1.27·7-s + 2.87·8-s + 9-s + 0.0671·12-s − 1.41·13-s − 1.77·14-s − 3.86·16-s + 5·17-s − 1.39·18-s − 0.158·19-s − 1.27·21-s + 5.00·23-s − 2.87·24-s + 1.97·26-s − 27-s − 0.0858·28-s − 6.27·29-s + 3.04·31-s − 0.379·32-s − 6.95·34-s − 0.0671·36-s + 4.69·37-s + 0.219·38-s + ⋯ |
L(s) = 1 | − 0.983·2-s − 0.577·3-s − 0.0335·4-s + 0.567·6-s + 0.482·7-s + 1.01·8-s + 0.333·9-s + 0.0193·12-s − 0.393·13-s − 0.474·14-s − 0.965·16-s + 1.21·17-s − 0.327·18-s − 0.0362·19-s − 0.278·21-s + 1.04·23-s − 0.586·24-s + 0.386·26-s − 0.192·27-s − 0.0162·28-s − 1.16·29-s + 0.547·31-s − 0.0671·32-s − 1.19·34-s − 0.0111·36-s + 0.772·37-s + 0.0356·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9335335275\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9335335275\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.39T + 2T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 0.158T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 - 4.69T + 37T^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 - 5.41T + 43T^{2} \) |
| 47 | \( 1 - 8.23T + 47T^{2} \) |
| 53 | \( 1 + 9.36T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 - 2.54T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 6.41T + 97T^{2} \) |
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
−7.88423651793036996497488756018, −7.20335030778027364645189998275, −6.58616967369366925928841381418, −5.48606175769055731956761256923, −5.14235275868494274256476061703, −4.32682068350137788157233973972, −3.51391767224307827685700162191, −2.33405648913572026949088745086, −1.35777041084300549110178890208, −0.63164232851126800776034212570,
0.63164232851126800776034212570, 1.35777041084300549110178890208, 2.33405648913572026949088745086, 3.51391767224307827685700162191, 4.32682068350137788157233973972, 5.14235275868494274256476061703, 5.48606175769055731956761256923, 6.58616967369366925928841381418, 7.20335030778027364645189998275, 7.88423651793036996497488756018