L(s) = 1 | + 1.41·2-s − 3·7-s − 2.82·8-s − 4.24·11-s − 3·13-s − 4.24·14-s − 4.00·16-s − 2.82·17-s + 19-s − 6·22-s + 7.07·23-s − 4.24·26-s + 4.24·29-s + 2·31-s − 4.00·34-s − 9·37-s + 1.41·38-s + 4.24·41-s − 6·43-s + 10.0·46-s − 2.82·47-s + 2·49-s − 9.89·53-s + 8.48·56-s + ⋯ |
L(s) = 1 | + 1.00·2-s − 1.13·7-s − 0.999·8-s − 1.27·11-s − 0.832·13-s − 1.13·14-s − 1.00·16-s − 0.685·17-s + 0.229·19-s − 1.27·22-s + 1.47·23-s − 0.832·26-s + 0.787·29-s + 0.359·31-s − 0.685·34-s − 1.47·37-s + 0.229·38-s + 0.662·41-s − 0.914·43-s + 1.47·46-s − 0.412·47-s + 0.285·49-s − 1.35·53-s + 1.13·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 9T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 9.89T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04254508366443694437296010122, −9.316019640774248329195426581042, −8.381780227806597238431155974937, −7.13177171038265727200749278672, −6.37413879572584210075541338248, −5.26620491602508933282131027767, −4.68147121582242178577071732276, −3.31542830773422409085022841573, −2.66331458192668103061953617832, 0,
2.66331458192668103061953617832, 3.31542830773422409085022841573, 4.68147121582242178577071732276, 5.26620491602508933282131027767, 6.37413879572584210075541338248, 7.13177171038265727200749278672, 8.381780227806597238431155974937, 9.316019640774248329195426581042, 10.04254508366443694437296010122