L(s) = 1 | − 2.57i·2-s − 4.64·4-s − 2.64·7-s + 6.82i·8-s − 0.913i·11-s + 6.82i·14-s + 8.29·16-s − 2.35·22-s + 9.39i·23-s + 12.2·28-s + 6.06i·29-s − 7.73i·32-s + 10.5·37-s − 5.29·43-s + 4.24i·44-s + ⋯ |
L(s) = 1 | − 1.82i·2-s − 2.32·4-s − 0.999·7-s + 2.41i·8-s − 0.275i·11-s + 1.82i·14-s + 2.07·16-s − 0.501·22-s + 1.95i·23-s + 2.32·28-s + 1.12i·29-s − 1.36i·32-s + 1.73·37-s − 0.806·43-s + 0.639i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8746286861\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8746286861\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 2 | \( 1 + 2.57iT - 2T^{2} \) |
| 11 | \( 1 + 0.913iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 9.39iT - 23T^{2} \) |
| 29 | \( 1 - 6.06iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14.5iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 7.57iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 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
−9.609133497703431134225628245679, −8.960354909094030442522124303824, −8.031550586256245359036309960950, −6.90361065606798613891948594130, −5.76379927290022681150116843764, −4.88865802796263248322511255460, −3.69595171713996581894541618111, −3.29524601271060746430909603846, −2.22011301085227501997125388417, −1.01324772106278416783974123480,
0.42818433567125402073333781455, 2.70742386067684719784832380015, 4.07729096095396247287052080441, 4.69083935676674260424930231941, 5.85220759204278267341394334634, 6.33530301830925455753161773921, 7.00513309847273813168451157713, 7.84291684097885515264896694460, 8.519265636390348719732774362429, 9.357224245587086380632108273598