L(s) = 1 | + 2.41·3-s − 1.89·5-s − 0.462·7-s + 2.81·9-s + 0.172·11-s + 0.314·13-s − 4.56·15-s − 7.25·17-s − 0.194·19-s − 1.11·21-s − 0.825·23-s − 1.41·25-s − 0.440·27-s + 3.71·29-s + 7.97·31-s + 0.415·33-s + 0.876·35-s − 6.20·37-s + 0.757·39-s − 1.31·41-s − 3.94·43-s − 5.33·45-s + 2.22·47-s − 6.78·49-s − 17.4·51-s + 10.0·53-s − 0.325·55-s + ⋯ |
L(s) = 1 | + 1.39·3-s − 0.846·5-s − 0.174·7-s + 0.939·9-s + 0.0518·11-s + 0.0871·13-s − 1.17·15-s − 1.75·17-s − 0.0445·19-s − 0.243·21-s − 0.172·23-s − 0.282·25-s − 0.0847·27-s + 0.689·29-s + 1.43·31-s + 0.0722·33-s + 0.148·35-s − 1.02·37-s + 0.121·39-s − 0.205·41-s − 0.601·43-s − 0.795·45-s + 0.324·47-s − 0.969·49-s − 2.44·51-s + 1.38·53-s − 0.0439·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + 1.89T + 5T^{2} \) |
| 7 | \( 1 + 0.462T + 7T^{2} \) |
| 11 | \( 1 - 0.172T + 11T^{2} \) |
| 13 | \( 1 - 0.314T + 13T^{2} \) |
| 17 | \( 1 + 7.25T + 17T^{2} \) |
| 19 | \( 1 + 0.194T + 19T^{2} \) |
| 23 | \( 1 + 0.825T + 23T^{2} \) |
| 29 | \( 1 - 3.71T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 + 6.20T + 37T^{2} \) |
| 41 | \( 1 + 1.31T + 41T^{2} \) |
| 43 | \( 1 + 3.94T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 6.02T + 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 9.68T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 9.75T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 8.75T + 89T^{2} \) |
| 97 | \( 1 - 9.75T + 97T^{2} \) |
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
−8.236745317357673416576472895395, −7.50948440804257496337662282983, −6.82839626322867221479796295085, −6.03317835605111554792149020505, −4.66018761253850194188973969474, −4.19705566364519988660733895348, −3.30212626280635608150194630592, −2.65595391378646636224193591514, −1.68219006718632468243267875047, 0,
1.68219006718632468243267875047, 2.65595391378646636224193591514, 3.30212626280635608150194630592, 4.19705566364519988660733895348, 4.66018761253850194188973969474, 6.03317835605111554792149020505, 6.82839626322867221479796295085, 7.50948440804257496337662282983, 8.236745317357673416576472895395