L(s) = 1 | − 3-s + 5-s − 2·9-s + 5·11-s + 13-s − 15-s + 3·17-s − 5·19-s − 3·23-s − 4·25-s + 5·27-s − 6·29-s − 3·31-s − 5·33-s − 3·37-s − 39-s + 6·41-s − 8·43-s − 2·45-s − 9·47-s − 3·51-s − 5·53-s + 5·55-s + 5·57-s + 59-s + 7·61-s + 65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s + 1.50·11-s + 0.277·13-s − 0.258·15-s + 0.727·17-s − 1.14·19-s − 0.625·23-s − 4/5·25-s + 0.962·27-s − 1.11·29-s − 0.538·31-s − 0.870·33-s − 0.493·37-s − 0.160·39-s + 0.937·41-s − 1.21·43-s − 0.298·45-s − 1.31·47-s − 0.420·51-s − 0.686·53-s + 0.674·55-s + 0.662·57-s + 0.130·59-s + 0.896·61-s + 0.124·65-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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.000180724788068425592034824699, −6.87794646655112694812765802913, −6.35842507273556307721675628008, −5.78264617786917595068787528824, −5.12228555607061517855069457512, −4.02848368131337530291632060593, −3.49737015913083891815811065712, −2.20654137035858210917480123322, −1.37828020940715961705077685788, 0,
1.37828020940715961705077685788, 2.20654137035858210917480123322, 3.49737015913083891815811065712, 4.02848368131337530291632060593, 5.12228555607061517855069457512, 5.78264617786917595068787528824, 6.35842507273556307721675628008, 6.87794646655112694812765802913, 8.000180724788068425592034824699