L(s) = 1 | + (0.707 − 0.707i)3-s + (0.334 + 0.334i)5-s + 4.55i·7-s − 1.00i·9-s + (2.47 + 2.47i)11-s + (0.0594 − 0.0594i)13-s + 0.473·15-s + 3.61·17-s + (−2.55 + 2.55i)19-s + (3.22 + 3.22i)21-s − 2.82i·23-s − 4.77i·25-s + (−0.707 − 0.707i)27-s + (5.16 − 5.16i)29-s − 0.557·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.149 + 0.149i)5-s + 1.72i·7-s − 0.333i·9-s + (0.745 + 0.745i)11-s + (0.0164 − 0.0164i)13-s + 0.122·15-s + 0.877·17-s + (−0.586 + 0.586i)19-s + (0.703 + 0.703i)21-s − 0.589i·23-s − 0.955i·25-s + (−0.136 − 0.136i)27-s + (0.958 − 0.958i)29-s − 0.100·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56350 + 0.408721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56350 + 0.408721i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (-0.334 - 0.334i)T + 5iT^{2} \) |
| 7 | \( 1 - 4.55iT - 7T^{2} \) |
| 11 | \( 1 + (-2.47 - 2.47i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.0594 + 0.0594i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 + (2.55 - 2.55i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-5.16 + 5.16i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.557T + 31T^{2} \) |
| 37 | \( 1 + (4.38 + 4.38i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.27iT - 41T^{2} \) |
| 43 | \( 1 + (-1.61 - 1.61i)T + 43iT^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-0.493 - 0.493i)T + 53iT^{2} \) |
| 59 | \( 1 + (4 + 4i)T + 59iT^{2} \) |
| 61 | \( 1 + (2.72 - 2.72i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.77 - 3.77i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.11iT - 71T^{2} \) |
| 73 | \( 1 - 0.541iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−11.84992528163165615995575595763, −10.34113250303791978491275644085, −9.453482378376616199073254043895, −8.651375874094775233556693767952, −7.85410705081533497271810187617, −6.51108940158708060113879518032, −5.85744070877236210604722731863, −4.47292682848476598746833523159, −2.91289612241523218119961145656, −1.89020047741944756437274000865,
1.20603024807497834306164677842, 3.30688755352605506775610469418, 4.07705397765460023138055660320, 5.27656954579541939383551991246, 6.69148333313821998587039406224, 7.49643283851883970303424838483, 8.583496898322148199759175567457, 9.455655516276754496600053485835, 10.45218349938692226375152381503, 10.95622444051876198825869411093