| L(s) = 1 | − 5·5-s − 23.9·7-s − 57.9·11-s − 8.16·13-s − 50.0·17-s − 69.7·19-s − 4.92·23-s + 25·25-s + 79.4·29-s − 260.·31-s + 119.·35-s − 223.·37-s − 337.·41-s − 326.·43-s − 89.6·47-s + 229.·49-s − 543.·53-s + 289.·55-s − 92·59-s + 159.·61-s + 40.8·65-s + 910.·67-s + 293.·71-s + 142.·73-s + 1.38e3·77-s − 1.10e3·79-s − 813.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.29·7-s − 1.58·11-s − 0.174·13-s − 0.714·17-s − 0.842·19-s − 0.0446·23-s + 0.200·25-s + 0.508·29-s − 1.50·31-s + 0.577·35-s − 0.994·37-s − 1.28·41-s − 1.15·43-s − 0.278·47-s + 0.668·49-s − 1.40·53-s + 0.710·55-s − 0.203·59-s + 0.334·61-s + 0.0778·65-s + 1.66·67-s + 0.490·71-s + 0.227·73-s + 2.05·77-s − 1.57·79-s − 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.06334128894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06334128894\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 23.9T + 343T^{2} \) |
| 11 | \( 1 + 57.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.16T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 4.92T + 1.21e4T^{2} \) |
| 29 | \( 1 - 79.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 260.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 89.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 543.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 92T + 2.05e5T^{2} \) |
| 61 | \( 1 - 159.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 910.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 293.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 813.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 956.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 106.T + 9.12e5T^{2} \) |
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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
−8.623617526308794824466521541341, −8.034063023759955261150848424966, −7.03314455088647665289434335523, −6.55699466174677142709572415380, −5.50374486558184156591473013830, −4.76160230446448533195050346214, −3.65140508153700652794686853834, −2.95361400622317892568025384428, −1.96972612622166197729846886069, −0.10528843580233186873871118968,
0.10528843580233186873871118968, 1.96972612622166197729846886069, 2.95361400622317892568025384428, 3.65140508153700652794686853834, 4.76160230446448533195050346214, 5.50374486558184156591473013830, 6.55699466174677142709572415380, 7.03314455088647665289434335523, 8.034063023759955261150848424966, 8.623617526308794824466521541341
