L(s) = 1 | − 2.06·2-s + 3.16·3-s + 2.27·4-s − 5-s − 6.55·6-s + 3.93·7-s − 0.566·8-s + 7.04·9-s + 2.06·10-s − 1.10·11-s + 7.20·12-s + 6.41·13-s − 8.14·14-s − 3.16·15-s − 3.37·16-s + 7.11·17-s − 14.5·18-s + 0.694·19-s − 2.27·20-s + 12.4·21-s + 2.28·22-s − 7.97·23-s − 1.79·24-s + 25-s − 13.2·26-s + 12.8·27-s + 8.95·28-s + ⋯ |
L(s) = 1 | − 1.46·2-s + 1.82·3-s + 1.13·4-s − 0.447·5-s − 2.67·6-s + 1.48·7-s − 0.200·8-s + 2.34·9-s + 0.653·10-s − 0.333·11-s + 2.08·12-s + 1.77·13-s − 2.17·14-s − 0.818·15-s − 0.844·16-s + 1.72·17-s − 3.43·18-s + 0.159·19-s − 0.508·20-s + 2.72·21-s + 0.487·22-s − 1.66·23-s − 0.366·24-s + 0.200·25-s − 2.60·26-s + 2.46·27-s + 1.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.817035085\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.817035085\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 3 | \( 1 - 3.16T + 3T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 6.41T + 13T^{2} \) |
| 17 | \( 1 - 7.11T + 17T^{2} \) |
| 19 | \( 1 - 0.694T + 19T^{2} \) |
| 23 | \( 1 + 7.97T + 23T^{2} \) |
| 29 | \( 1 + 9.53T + 29T^{2} \) |
| 31 | \( 1 + 7.06T + 31T^{2} \) |
| 37 | \( 1 + 5.67T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 - 1.99T + 47T^{2} \) |
| 53 | \( 1 + 6.08T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 - 6.40T + 61T^{2} \) |
| 67 | \( 1 - 0.552T + 67T^{2} \) |
| 71 | \( 1 - 0.706T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 + 9.29T + 79T^{2} \) |
| 83 | \( 1 - 5.06T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 5.95T + 97T^{2} \) |
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
−9.452744638380767986976617865016, −8.705597639853358993633786133505, −8.193592403609000849952205958072, −7.78876663608133069626958397316, −7.25350960512593102451211291508, −5.58567311054620816306553887257, −4.11715777862299849463197895265, −3.44783773218946711948490549299, −1.93086462973735844580330102084, −1.39856020950431869369959229008,
1.39856020950431869369959229008, 1.93086462973735844580330102084, 3.44783773218946711948490549299, 4.11715777862299849463197895265, 5.58567311054620816306553887257, 7.25350960512593102451211291508, 7.78876663608133069626958397316, 8.193592403609000849952205958072, 8.705597639853358993633786133505, 9.452744638380767986976617865016