| L(s) = 1 | + 3.15·3-s + 0.0614·5-s − 2.55·7-s + 6.95·9-s − 4.48·11-s − 1.07·13-s + 0.193·15-s − 3.56·17-s − 5.69·19-s − 8.05·21-s − 4.99·25-s + 12.4·27-s + 5.05·29-s − 9.56·31-s − 14.1·33-s − 0.156·35-s + 6.40·37-s − 3.38·39-s − 0.418·41-s + 1.35·43-s + 0.427·45-s − 9.28·47-s − 0.486·49-s − 11.2·51-s + 2.78·53-s − 0.275·55-s − 17.9·57-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 0.0274·5-s − 0.964·7-s + 2.31·9-s − 1.35·11-s − 0.297·13-s + 0.0500·15-s − 0.864·17-s − 1.30·19-s − 1.75·21-s − 0.999·25-s + 2.40·27-s + 0.939·29-s − 1.71·31-s − 2.46·33-s − 0.0265·35-s + 1.05·37-s − 0.542·39-s − 0.0653·41-s + 0.205·43-s + 0.0637·45-s − 1.35·47-s − 0.0695·49-s − 1.57·51-s + 0.382·53-s − 0.0371·55-s − 2.37·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 - 0.0614T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 + 1.07T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 5.69T + 19T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 + 9.56T + 31T^{2} \) |
| 37 | \( 1 - 6.40T + 37T^{2} \) |
| 41 | \( 1 + 0.418T + 41T^{2} \) |
| 43 | \( 1 - 1.35T + 43T^{2} \) |
| 47 | \( 1 + 9.28T + 47T^{2} \) |
| 53 | \( 1 - 2.78T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 6.99T + 67T^{2} \) |
| 71 | \( 1 + 0.204T + 71T^{2} \) |
| 73 | \( 1 - 0.687T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 6.90T + 97T^{2} \) |
| 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
−8.181336799860494237130950949344, −7.46269814789873205935089576953, −6.81251755732835517255221363685, −5.95595847177167420799660776853, −4.76505938336227211998373473799, −4.02680948590652182708987504075, −3.21307396059986957256067328261, −2.51625262369634643070669571416, −1.93051844814768305172406664986, 0,
1.93051844814768305172406664986, 2.51625262369634643070669571416, 3.21307396059986957256067328261, 4.02680948590652182708987504075, 4.76505938336227211998373473799, 5.95595847177167420799660776853, 6.81251755732835517255221363685, 7.46269814789873205935089576953, 8.181336799860494237130950949344
