| L(s) = 1 | − 2·2-s + 4·4-s + 21·5-s + 8·7-s − 8·8-s − 42·10-s + 36·11-s − 49·13-s − 16·14-s + 16·16-s + 21·17-s − 112·19-s + 84·20-s − 72·22-s + 180·23-s + 316·25-s + 98·26-s + 32·28-s − 135·29-s + 308·31-s − 32·32-s − 42·34-s + 168·35-s − 37-s + 224·38-s − 168·40-s − 42·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.87·5-s + 0.431·7-s − 0.353·8-s − 1.32·10-s + 0.986·11-s − 1.04·13-s − 0.305·14-s + 1/4·16-s + 0.299·17-s − 1.35·19-s + 0.939·20-s − 0.697·22-s + 1.63·23-s + 2.52·25-s + 0.739·26-s + 0.215·28-s − 0.864·29-s + 1.78·31-s − 0.176·32-s − 0.211·34-s + 0.811·35-s − 0.00444·37-s + 0.956·38-s − 0.664·40-s − 0.159·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.778422857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.778422857\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 21 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 49 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 + 112 T + p^{3} T^{2} \) |
| 23 | \( 1 - 180 T + p^{3} T^{2} \) |
| 29 | \( 1 + 135 T + p^{3} T^{2} \) |
| 31 | \( 1 - 308 T + p^{3} T^{2} \) |
| 37 | \( 1 + T + p^{3} T^{2} \) |
| 41 | \( 1 + 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 20 T + p^{3} T^{2} \) |
| 47 | \( 1 - 84 T + p^{3} T^{2} \) |
| 53 | \( 1 + 174 T + p^{3} T^{2} \) |
| 59 | \( 1 - 504 T + p^{3} T^{2} \) |
| 61 | \( 1 + 385 T + p^{3} T^{2} \) |
| 67 | \( 1 - 272 T + p^{3} T^{2} \) |
| 71 | \( 1 + 888 T + p^{3} T^{2} \) |
| 73 | \( 1 - 371 T + p^{3} T^{2} \) |
| 79 | \( 1 + 652 T + p^{3} T^{2} \) |
| 83 | \( 1 - 84 T + p^{3} T^{2} \) |
| 89 | \( 1 - 21 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1246 T + p^{3} T^{2} \) |
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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
−12.41394071236403716855235201174, −11.13653355684202611096398594344, −10.14083187295435965369719572504, −9.429598107768279837588045833296, −8.589465538681456384087449286375, −7.00279410106358566342457441488, −6.13723930276088316962253176521, −4.88273130016418032510317712503, −2.56064079514262247290916296390, −1.38591109412442047027913820066,
1.38591109412442047027913820066, 2.56064079514262247290916296390, 4.88273130016418032510317712503, 6.13723930276088316962253176521, 7.00279410106358566342457441488, 8.589465538681456384087449286375, 9.429598107768279837588045833296, 10.14083187295435965369719572504, 11.13653355684202611096398594344, 12.41394071236403716855235201174
