| L(s) = 1 | − 3·3-s + 2·5-s − 7·7-s + 9·9-s − 52·11-s − 86·13-s − 6·15-s − 30·17-s + 4·19-s + 21·21-s + 120·23-s − 121·25-s − 27·27-s − 246·29-s + 80·31-s + 156·33-s − 14·35-s + 290·37-s + 258·39-s − 374·41-s − 164·43-s + 18·45-s + 464·47-s + 49·49-s + 90·51-s + 162·53-s − 104·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.178·5-s − 0.377·7-s + 1/3·9-s − 1.42·11-s − 1.83·13-s − 0.103·15-s − 0.428·17-s + 0.0482·19-s + 0.218·21-s + 1.08·23-s − 0.967·25-s − 0.192·27-s − 1.57·29-s + 0.463·31-s + 0.822·33-s − 0.0676·35-s + 1.28·37-s + 1.05·39-s − 1.42·41-s − 0.581·43-s + 0.0596·45-s + 1.44·47-s + 1/7·49-s + 0.247·51-s + 0.419·53-s − 0.254·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6589802860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6589802860\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 246 T + p^{3} T^{2} \) |
| 31 | \( 1 - 80 T + p^{3} T^{2} \) |
| 37 | \( 1 - 290 T + p^{3} T^{2} \) |
| 41 | \( 1 + 374 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 464 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 180 T + p^{3} T^{2} \) |
| 61 | \( 1 - 666 T + p^{3} T^{2} \) |
| 67 | \( 1 - 628 T + p^{3} T^{2} \) |
| 71 | \( 1 - 296 T + p^{3} T^{2} \) |
| 73 | \( 1 + 518 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1184 T + p^{3} T^{2} \) |
| 83 | \( 1 + 220 T + p^{3} T^{2} \) |
| 89 | \( 1 + 774 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 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
−9.582466104581923338549598791230, −8.356035654985988535393463337988, −7.41002763873039836593099024644, −6.95287653253057540982628809989, −5.69178870482611688250403523880, −5.20835047444486238345524639061, −4.28803873691617538786777764382, −2.90538436207715549795212512791, −2.09358513811269973538901145435, −0.39217243862332876031574110472,
0.39217243862332876031574110472, 2.09358513811269973538901145435, 2.90538436207715549795212512791, 4.28803873691617538786777764382, 5.20835047444486238345524639061, 5.69178870482611688250403523880, 6.95287653253057540982628809989, 7.41002763873039836593099024644, 8.356035654985988535393463337988, 9.582466104581923338549598791230
