L(s) = 1 | + 3-s + 4·5-s − 4·7-s + 9-s − 2·13-s + 4·15-s + 4·17-s − 4·19-s − 4·21-s + 11·25-s + 27-s + 2·29-s + 8·31-s − 16·35-s − 8·37-s − 2·39-s − 6·41-s − 4·43-s + 4·45-s + 9·49-s + 4·51-s − 4·53-s − 4·57-s + 4·59-s + 8·61-s − 4·63-s − 8·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s + 1.03·15-s + 0.970·17-s − 0.917·19-s − 0.872·21-s + 11/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 2.70·35-s − 1.31·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.596·45-s + 9/7·49-s + 0.560·51-s − 0.549·53-s − 0.529·57-s + 0.520·59-s + 1.02·61-s − 0.503·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.313328003\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.313328003\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.27936533499681, −14.69458066928420, −14.19205442207454, −13.66694214364442, −13.30381042603225, −12.82710233566109, −12.32252359117908, −11.78034816089344, −10.58674650353102, −10.13049095872470, −10.01542138889235, −9.441756305644795, −8.829344856431413, −8.402751174820727, −7.437256841210951, −6.748391172496067, −6.394270545291348, −5.870248617730541, −5.155242074435087, −4.522568687838610, −3.380571914542803, −3.085761730606662, −2.293576704311719, −1.744224133817025, −0.6745169486398542,
0.6745169486398542, 1.744224133817025, 2.293576704311719, 3.085761730606662, 3.380571914542803, 4.522568687838610, 5.155242074435087, 5.870248617730541, 6.394270545291348, 6.748391172496067, 7.437256841210951, 8.402751174820727, 8.829344856431413, 9.441756305644795, 10.01542138889235, 10.13049095872470, 10.58674650353102, 11.78034816089344, 12.32252359117908, 12.82710233566109, 13.30381042603225, 13.66694214364442, 14.19205442207454, 14.69458066928420, 15.27936533499681