L(s) = 1 | − 3-s + 3·5-s − 2·9-s − 11-s − 13-s − 3·15-s − 5·17-s − 3·19-s + 9·23-s + 4·25-s + 5·27-s − 6·29-s + 7·31-s + 33-s + 7·37-s + 39-s + 6·41-s − 12·43-s − 6·45-s − 7·47-s + 5·51-s − 9·53-s − 3·55-s + 3·57-s + 3·59-s − 5·61-s − 3·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s − 2/3·9-s − 0.301·11-s − 0.277·13-s − 0.774·15-s − 1.21·17-s − 0.688·19-s + 1.87·23-s + 4/5·25-s + 0.962·27-s − 1.11·29-s + 1.25·31-s + 0.174·33-s + 1.15·37-s + 0.160·39-s + 0.937·41-s − 1.82·43-s − 0.894·45-s − 1.02·47-s + 0.700·51-s − 1.23·53-s − 0.404·55-s + 0.397·57-s + 0.390·59-s − 0.640·61-s − 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.929778722940787664897183775290, −6.77504545398848891877312802636, −6.45313995450814007271080542757, −5.69487992925904129865157019434, −5.06453608707591247061619946062, −4.43711428360969900198644812030, −2.99959448466144951655638035592, −2.42639622511245724328489397026, −1.39372934816215455298377143298, 0,
1.39372934816215455298377143298, 2.42639622511245724328489397026, 2.99959448466144951655638035592, 4.43711428360969900198644812030, 5.06453608707591247061619946062, 5.69487992925904129865157019434, 6.45313995450814007271080542757, 6.77504545398848891877312802636, 7.929778722940787664897183775290