L(s) = 1 | + 3-s + 5-s + 3.30·7-s + 9-s − 2.24·11-s − 3.19·13-s + 15-s + 3.55·17-s + 19-s + 3.30·21-s − 1.55·23-s + 25-s + 27-s + 4.24·29-s + 7.43·31-s − 2.24·33-s + 3.30·35-s − 5.30·37-s − 3.19·39-s + 2.36·41-s − 3.80·43-s + 45-s + 9.55·47-s + 3.94·49-s + 3.55·51-s − 3.55·53-s − 2.24·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.25·7-s + 0.333·9-s − 0.678·11-s − 0.884·13-s + 0.258·15-s + 0.862·17-s + 0.229·19-s + 0.721·21-s − 0.324·23-s + 0.200·25-s + 0.192·27-s + 0.789·29-s + 1.33·31-s − 0.391·33-s + 0.559·35-s − 0.872·37-s − 0.510·39-s + 0.369·41-s − 0.580·43-s + 0.149·45-s + 1.39·47-s + 0.563·49-s + 0.498·51-s − 0.488·53-s − 0.303·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.094690156\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.094690156\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 7.43T + 31T^{2} \) |
| 37 | \( 1 + 5.30T + 37T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 + 3.80T + 43T^{2} \) |
| 47 | \( 1 - 9.55T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 + 0.498T + 59T^{2} \) |
| 61 | \( 1 + 1.17T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 0.117T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 7.68T + 97T^{2} \) |
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
−8.117338368412088124412081920818, −7.85847854663916155851515729755, −7.05323430166268022863876290340, −6.11913622713502240646875167457, −5.05972187290822889818492981446, −4.89613470246475703920396556132, −3.73646607515698203792198350918, −2.70171575211544791274023813309, −2.07185647403809080253698421841, −0.998472892298478562306170931813,
0.998472892298478562306170931813, 2.07185647403809080253698421841, 2.70171575211544791274023813309, 3.73646607515698203792198350918, 4.89613470246475703920396556132, 5.05972187290822889818492981446, 6.11913622713502240646875167457, 7.05323430166268022863876290340, 7.85847854663916155851515729755, 8.117338368412088124412081920818