L(s) = 1 | + i·2-s + (−0.707 + 0.707i)3-s − 4-s + (1.11 + 1.94i)5-s + (−0.707 − 0.707i)6-s + (0.636 − 0.636i)7-s − i·8-s − 1.00i·9-s + (−1.94 + 1.11i)10-s − 2.26i·11-s + (0.707 − 0.707i)12-s + 1.29i·13-s + (0.636 + 0.636i)14-s + (−2.15 − 0.586i)15-s + 16-s − 5.74·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (0.496 + 0.867i)5-s + (−0.288 − 0.288i)6-s + (0.240 − 0.240i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.613 + 0.351i)10-s − 0.684i·11-s + (0.204 − 0.204i)12-s + 0.358i·13-s + (0.170 + 0.170i)14-s + (−0.557 − 0.151i)15-s + 0.250·16-s − 1.39·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7637147887\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7637147887\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.11 - 1.94i)T \) |
| 37 | \( 1 + (-5.19 + 3.16i)T \) |
good | 7 | \( 1 + (-0.636 + 0.636i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.26iT - 11T^{2} \) |
| 13 | \( 1 - 1.29iT - 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 + (4.10 - 4.10i)T - 19iT^{2} \) |
| 23 | \( 1 - 8.53iT - 23T^{2} \) |
| 29 | \( 1 + (1.70 + 1.70i)T + 29iT^{2} \) |
| 31 | \( 1 + (5.81 - 5.81i)T - 31iT^{2} \) |
| 41 | \( 1 + 9.02iT - 41T^{2} \) |
| 43 | \( 1 - 0.853iT - 43T^{2} \) |
| 47 | \( 1 + (0.326 - 0.326i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.79 + 6.79i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.51 + 3.51i)T - 59iT^{2} \) |
| 61 | \( 1 + (2.06 - 2.06i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3.98 - 3.98i)T + 67iT^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + (11.8 - 11.8i)T - 73iT^{2} \) |
| 79 | \( 1 + (-9.04 + 9.04i)T - 79iT^{2} \) |
| 83 | \( 1 + (-9.12 - 9.12i)T + 83iT^{2} \) |
| 89 | \( 1 + (6.32 + 6.32i)T + 89iT^{2} \) |
| 97 | \( 1 - 9.28T + 97T^{2} \) |
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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
−10.33767505307435472664910302793, −9.407168299168708144897171418309, −8.763457002193468911372457247607, −7.64007068301927259803896103628, −6.88288256924893834540897404539, −6.06526612780173410656779240187, −5.47671297642807104334377889013, −4.26248141806560258373368655521, −3.44214753839720485336137234619, −1.87390911993904624268279592951,
0.33749020159285373491660657624, 1.82270449456122071460864907250, 2.55787752020733768046123466278, 4.47115907586272892984806832560, 4.70522425803776006751209081297, 5.93472278028792711676577530424, 6.70813490976889192236658971881, 7.928827970001401536072696498624, 8.761534837942006378271356909759, 9.314043929632035067649748881116