| L(s) = 1 | + 2.23·2-s − 2.99·4-s − 24.5·8-s − 31.0·16-s − 138.·17-s − 164·19-s + 98.3·23-s − 232·31-s + 127.·32-s − 310.·34-s − 366.·38-s + 220.·46-s + 545.·47-s − 343·49-s + 621.·53-s − 358·61-s − 518.·62-s + 533·64-s + 415.·68-s + 491.·76-s + 304·79-s − 1.27e3·83-s − 295.·92-s + 1.22e3·94-s − 766.·98-s + 1.39e3·106-s − 17.8·107-s + ⋯ |
| L(s) = 1 | + 0.790·2-s − 0.374·4-s − 1.08·8-s − 0.484·16-s − 1.97·17-s − 1.98·19-s + 0.891·23-s − 1.34·31-s + 0.704·32-s − 1.56·34-s − 1.56·38-s + 0.705·46-s + 1.69·47-s − 49-s + 1.61·53-s − 0.751·61-s − 1.06·62-s + 1.04·64-s + 0.741·68-s + 0.742·76-s + 0.432·79-s − 1.67·83-s − 0.334·92-s + 1.33·94-s − 0.790·98-s + 1.27·106-s − 0.0161·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 8T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.19e3T^{2} \) |
| 17 | \( 1 + 138.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 164T + 6.85e3T^{2} \) |
| 23 | \( 1 - 98.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 232T + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 - 545.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 621.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 358T + 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 - 304T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−11.38384544889922558773300890374, −10.55833654057932124804499891219, −9.092320934900559766495102421100, −8.621951082404838338011289046138, −6.97541723465667528651000855303, −6.00312121854706366471149544898, −4.74787374798103495310982545876, −3.93653285319028387175081764297, −2.36746407613640204428682890090, 0,
2.36746407613640204428682890090, 3.93653285319028387175081764297, 4.74787374798103495310982545876, 6.00312121854706366471149544898, 6.97541723465667528651000855303, 8.621951082404838338011289046138, 9.092320934900559766495102421100, 10.55833654057932124804499891219, 11.38384544889922558773300890374
