L(s) = 1 | + 3-s − 3.41·5-s − 2.82·7-s + 9-s − 2·11-s − 2.82·13-s − 3.41·15-s − 5.41·17-s − 1.17·19-s − 2.82·21-s − 5.65·23-s + 6.65·25-s + 27-s + 2·29-s − 6.82·31-s − 2·33-s + 9.65·35-s − 4·37-s − 2.82·39-s − 5.41·41-s − 0.242·43-s − 3.41·45-s + 12.8·47-s + 1.00·49-s − 5.41·51-s − 6.24·53-s + 6.82·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.52·5-s − 1.06·7-s + 0.333·9-s − 0.603·11-s − 0.784·13-s − 0.881·15-s − 1.31·17-s − 0.268·19-s − 0.617·21-s − 1.17·23-s + 1.33·25-s + 0.192·27-s + 0.371·29-s − 1.22·31-s − 0.348·33-s + 1.63·35-s − 0.657·37-s − 0.452·39-s − 0.845·41-s − 0.0370·43-s − 0.508·45-s + 1.87·47-s + 0.142·49-s − 0.758·51-s − 0.857·53-s + 0.920·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2522644775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2522644775\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 + 0.242T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 6.24T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 - 0.343T + 83T^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−7.74838956013818935383199512692, −7.27436743993728353419285787670, −6.70086571582468789286236077853, −5.82324893304271180834739113115, −4.75270594021407661630319812680, −4.17960514483653865191517050638, −3.51114989696741108455619800940, −2.83180567757970062369705848841, −1.98359931598991980555102016161, −0.22305366778578173736007627395,
0.22305366778578173736007627395, 1.98359931598991980555102016161, 2.83180567757970062369705848841, 3.51114989696741108455619800940, 4.17960514483653865191517050638, 4.75270594021407661630319812680, 5.82324893304271180834739113115, 6.70086571582468789286236077853, 7.27436743993728353419285787670, 7.74838956013818935383199512692