| L(s) = 1 | + 0.267·5-s − 0.464·13-s + 2.26·17-s − 4.92·25-s + 6.66·29-s + 11.3·37-s + 8·41-s − 7·49-s + 4·53-s − 5.39·61-s − 0.124·65-s + 10.8·73-s + 0.607·85-s + 16.6·89-s − 18·97-s + 20·101-s + 20.3·109-s + 20.1·113-s + ⋯ |
| L(s) = 1 | + 0.119·5-s − 0.128·13-s + 0.550·17-s − 0.985·25-s + 1.23·29-s + 1.87·37-s + 1.24·41-s − 49-s + 0.549·53-s − 0.690·61-s − 0.0154·65-s + 1.27·73-s + 0.0659·85-s + 1.76·89-s − 1.82·97-s + 1.99·101-s + 1.94·109-s + 1.89·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.847169221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.847169221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.267T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.464T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−8.924617339451835593653033504269, −7.987305659786083206758074514462, −7.53993379307156996093139778832, −6.44488767609829650456319452562, −5.89902613717607467979422048431, −4.92110471001131665974461053655, −4.13401481753795826376455251387, −3.11397351229638985652968555255, −2.16936907008936543068345935141, −0.870512331980832436414106425658,
0.870512331980832436414106425658, 2.16936907008936543068345935141, 3.11397351229638985652968555255, 4.13401481753795826376455251387, 4.92110471001131665974461053655, 5.89902613717607467979422048431, 6.44488767609829650456319452562, 7.53993379307156996093139778832, 7.987305659786083206758074514462, 8.924617339451835593653033504269
