| L(s) = 1 | − 0.582·3-s − 3.18·5-s − 3.00·7-s − 2.66·9-s + 2.52·11-s + 2.23·13-s + 1.85·15-s + 4.66·17-s − 4.31·19-s + 1.75·21-s + 4.90·23-s + 5.17·25-s + 3.29·27-s + 8.71·29-s − 1.05·31-s − 1.47·33-s + 9.59·35-s − 5.94·37-s − 1.29·39-s − 3.55·41-s + 4.35·43-s + 8.48·45-s + 2.89·47-s + 2.04·49-s − 2.71·51-s − 1.27·53-s − 8.05·55-s + ⋯ |
| L(s) = 1 | − 0.336·3-s − 1.42·5-s − 1.13·7-s − 0.886·9-s + 0.761·11-s + 0.618·13-s + 0.479·15-s + 1.13·17-s − 0.990·19-s + 0.382·21-s + 1.02·23-s + 1.03·25-s + 0.634·27-s + 1.61·29-s − 0.188·31-s − 0.256·33-s + 1.62·35-s − 0.977·37-s − 0.208·39-s − 0.555·41-s + 0.663·43-s + 1.26·45-s + 0.422·47-s + 0.292·49-s − 0.380·51-s − 0.174·53-s − 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.582T + 3T^{2} \) |
| 5 | \( 1 + 3.18T + 5T^{2} \) |
| 7 | \( 1 + 3.00T + 7T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 - 4.66T + 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 23 | \( 1 - 4.90T + 23T^{2} \) |
| 29 | \( 1 - 8.71T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 + 5.94T + 37T^{2} \) |
| 41 | \( 1 + 3.55T + 41T^{2} \) |
| 43 | \( 1 - 4.35T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 59 | \( 1 + 0.00300T + 59T^{2} \) |
| 61 | \( 1 - 5.70T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 2.05T + 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 0.508T + 83T^{2} \) |
| 89 | \( 1 + 7.67T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
| 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
−8.222162160725532416648435753480, −7.24926457762140455313146017493, −6.59499988813711744630189071465, −6.02638071425345771740477426153, −5.05234388909847729831100622920, −4.09357806677722359320381276807, −3.43161448870414746699678477875, −2.84621727413788905260282510715, −1.05090966686963273835519337678, 0,
1.05090966686963273835519337678, 2.84621727413788905260282510715, 3.43161448870414746699678477875, 4.09357806677722359320381276807, 5.05234388909847729831100622920, 6.02638071425345771740477426153, 6.59499988813711744630189071465, 7.24926457762140455313146017493, 8.222162160725532416648435753480
