| L(s) = 1 | + (−0.228 + 1.39i)2-s + (−1.89 − 0.637i)4-s − 2.64·7-s + (1.32 − 2.49i)8-s + 3·9-s + 6.10i·11-s + (0.604 − 3.69i)14-s + (3.18 + 2.41i)16-s + (−0.685 + 4.18i)18-s + (−8.52 − 1.39i)22-s − 9.57·23-s + (5.01 + 1.68i)28-s − 10.1·29-s + (−4.10 + 3.89i)32-s + (−5.68 − 1.91i)36-s + 6.16i·37-s + ⋯ |
| L(s) = 1 | + (−0.161 + 0.986i)2-s + (−0.947 − 0.318i)4-s − 0.999·7-s + (0.467 − 0.883i)8-s + 9-s + 1.84i·11-s + (0.161 − 0.986i)14-s + (0.796 + 0.604i)16-s + (−0.161 + 0.986i)18-s + (−1.81 − 0.297i)22-s − 1.99·23-s + (0.947 + 0.318i)28-s − 1.88·29-s + (−0.725 + 0.688i)32-s + (−0.947 − 0.318i)36-s + 1.01i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0452905 - 0.649749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0452905 - 0.649749i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.228 - 1.39i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 - 6.10iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 9.57T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.16iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 7.74T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 1.00iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.36136469594716063758702789292, −9.796191420257665768330568876033, −9.409841466716543572598520981563, −8.068720590217559642501153934244, −7.25488760479453382710260395012, −6.71016858666667845074444013049, −5.70000178255532697633252002002, −4.52471427345687762314277399440, −3.81034031616985835584705886749, −1.84174493706659170452486665831,
0.35718871683639559775363160860, 2.00169343661955720643783404275, 3.45541695835005750346648284906, 3.90407630548648826659765531099, 5.43000605412689274086387614339, 6.31712668376190556440832318080, 7.60343769339199376719140416141, 8.478206636750622722088430499320, 9.401636824284219619240233198526, 10.00818701027628551977852412929
