| L(s) = 1 | + 2.68i·2-s − 5.19·4-s − i·7-s − 8.56i·8-s − 3.19·11-s − 6.38i·13-s + 2.68·14-s + 12.5·16-s + 5.36i·17-s + 2.38·19-s − 8.57i·22-s + 3.19i·23-s + 17.1·26-s + 5.19i·28-s + 2.16·29-s + ⋯ |
| L(s) = 1 | + 1.89i·2-s − 2.59·4-s − 0.377i·7-s − 3.02i·8-s − 0.964·11-s − 1.77i·13-s + 0.716·14-s + 3.14·16-s + 1.30i·17-s + 0.547·19-s − 1.82i·22-s + 0.666i·23-s + 3.35·26-s + 0.981i·28-s + 0.402·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.220936359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.220936359\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 2 | \( 1 - 2.68iT - 2T^{2} \) |
| 11 | \( 1 + 3.19T + 11T^{2} \) |
| 13 | \( 1 + 6.38iT - 13T^{2} \) |
| 17 | \( 1 - 5.36iT - 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 - 3.19iT - 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 9.38iT - 43T^{2} \) |
| 47 | \( 1 + 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 9.38iT - 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 4.38iT - 73T^{2} \) |
| 79 | \( 1 - 5.38T + 79T^{2} \) |
| 83 | \( 1 - 4.33iT - 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 + 10.7iT - 97T^{2} \) |
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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
−9.520688211638064684268419671619, −8.287410187993557078264063335731, −8.063154172298419596265075850277, −7.40956730348986479087947651000, −6.41817772094764414493657220948, −5.68421777018389364642466114739, −5.12361313190500976648048853927, −4.12613832004756132883808304498, −3.06541704021114377646044966162, −0.76174286644347241860624873776,
0.800552624707541126172825028039, 2.24535447504166971945491152603, 2.69118405787036053489710805693, 3.91835069558195567804720340168, 4.70533612438452772791044110887, 5.40689710899919309390409809270, 6.76846148760668283818574590041, 7.920942347875284339646683287527, 8.820216688348123050562566022401, 9.409226618434673982951930035206
