L(s) = 1 | − 3-s − 7-s − 2·9-s + 3·13-s + 7·17-s + 19-s + 21-s − 5·23-s + 5·27-s − 5·29-s − 10·31-s − 2·37-s − 3·39-s + 2·41-s + 6·43-s − 6·49-s − 7·51-s − 9·53-s − 57-s + 7·59-s − 4·61-s + 2·63-s + 7·67-s + 5·69-s + 9·73-s + 10·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.832·13-s + 1.69·17-s + 0.229·19-s + 0.218·21-s − 1.04·23-s + 0.962·27-s − 0.928·29-s − 1.79·31-s − 0.328·37-s − 0.480·39-s + 0.312·41-s + 0.914·43-s − 6/7·49-s − 0.980·51-s − 1.23·53-s − 0.132·57-s + 0.911·59-s − 0.512·61-s + 0.251·63-s + 0.855·67-s + 0.601·69-s + 1.05·73-s + 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.61807435785284943275541723093, −6.73197936816745076811971284882, −5.85904685656139178134016938288, −5.71882894378019900550294615571, −4.88170521322202746529885574091, −3.64026094603493587105007789328, −3.42755378925346026650881759464, −2.19725007951254921850720539408, −1.13723262639258930019139286176, 0,
1.13723262639258930019139286176, 2.19725007951254921850720539408, 3.42755378925346026650881759464, 3.64026094603493587105007789328, 4.88170521322202746529885574091, 5.71882894378019900550294615571, 5.85904685656139178134016938288, 6.73197936816745076811971284882, 7.61807435785284943275541723093