| L(s) = 1 | − 2-s + 2.48·3-s + 4-s + 5-s − 2.48·6-s − 0.431·7-s − 8-s + 3.18·9-s − 10-s + 2.48·12-s + 3.94·13-s + 0.431·14-s + 2.48·15-s + 16-s + 3.42·17-s − 3.18·18-s − 6.12·19-s + 20-s − 1.07·21-s + 4.17·23-s − 2.48·24-s + 25-s − 3.94·26-s + 0.462·27-s − 0.431·28-s − 0.633·29-s − 2.48·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.43·3-s + 0.5·4-s + 0.447·5-s − 1.01·6-s − 0.163·7-s − 0.353·8-s + 1.06·9-s − 0.316·10-s + 0.717·12-s + 1.09·13-s + 0.115·14-s + 0.642·15-s + 0.250·16-s + 0.829·17-s − 0.750·18-s − 1.40·19-s + 0.223·20-s − 0.234·21-s + 0.869·23-s − 0.507·24-s + 0.200·25-s − 0.773·26-s + 0.0890·27-s − 0.0816·28-s − 0.117·29-s − 0.454·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.200914645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200914645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.48T + 3T^{2} \) |
| 7 | \( 1 + 0.431T + 7T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 19 | \( 1 + 6.12T + 19T^{2} \) |
| 23 | \( 1 - 4.17T + 23T^{2} \) |
| 29 | \( 1 + 0.633T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 + 5.24T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 7.44T + 79T^{2} \) |
| 83 | \( 1 - 7.88T + 83T^{2} \) |
| 89 | \( 1 - 1.34T + 89T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
| 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
−9.562061959366196850241268342924, −8.727430650957411342079178459689, −8.458035978539003794269081425429, −7.54338628332855038966989334739, −6.61677055421742821444760022268, −5.76302549809444576068258591231, −4.27151026041002070681544902976, −3.23726686724881863091563156210, −2.43618743311062742377272800178, −1.28927567059979957327776163327,
1.28927567059979957327776163327, 2.43618743311062742377272800178, 3.23726686724881863091563156210, 4.27151026041002070681544902976, 5.76302549809444576068258591231, 6.61677055421742821444760022268, 7.54338628332855038966989334739, 8.458035978539003794269081425429, 8.727430650957411342079178459689, 9.562061959366196850241268342924
