| L(s) = 1 | + (0.891 + 0.453i)2-s + (1.61 − 0.256i)3-s + (0.587 + 0.809i)4-s + (−2.22 + 0.182i)5-s + (1.55 + 0.506i)6-s + (0.119 − 0.753i)7-s + (0.156 + 0.987i)8-s + (−0.299 + 0.0972i)9-s + (−2.06 − 0.849i)10-s + (−1.49 − 2.95i)11-s + (1.15 + 1.15i)12-s + (−0.387 + 0.760i)13-s + (0.448 − 0.616i)14-s + (−3.56 + 0.866i)15-s + (−0.309 + 0.951i)16-s + (−0.657 − 1.29i)17-s + ⋯ |
| L(s) = 1 | + (0.630 + 0.321i)2-s + (0.934 − 0.148i)3-s + (0.293 + 0.404i)4-s + (−0.996 + 0.0816i)5-s + (0.636 + 0.206i)6-s + (0.0450 − 0.284i)7-s + (0.0553 + 0.349i)8-s + (−0.0997 + 0.0324i)9-s + (−0.654 − 0.268i)10-s + (−0.451 − 0.892i)11-s + (0.334 + 0.334i)12-s + (−0.107 + 0.210i)13-s + (0.119 − 0.164i)14-s + (−0.919 + 0.223i)15-s + (−0.0772 + 0.237i)16-s + (−0.159 − 0.312i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52300 + 0.262085i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52300 + 0.262085i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.891 - 0.453i)T \) |
| 5 | \( 1 + (2.22 - 0.182i)T \) |
| 11 | \( 1 + (1.49 + 2.95i)T \) |
| good | 3 | \( 1 + (-1.61 + 0.256i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (-0.119 + 0.753i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (0.387 - 0.760i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.657 + 1.29i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-4.09 - 2.97i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.65 - 1.65i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.552 + 0.401i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.08 + 3.34i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (8.01 + 1.26i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-7.30 + 10.0i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-7.61 - 7.61i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.98 - 12.5i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (7.31 + 3.72i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (0.254 + 0.350i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-12.0 - 3.91i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (3.82 + 3.82i)T + 67iT^{2} \) |
| 71 | \( 1 + (2.45 - 7.55i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.13 - 0.179i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (3.11 + 9.59i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.212 - 0.108i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + 11.2iT - 89T^{2} \) |
| 97 | \( 1 + (-6.45 + 12.6i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−14.05349306853106953199752549833, −12.90657898115069734579676616096, −11.76621292330904387414542743563, −10.84859188209757353918623089263, −9.107094561953468903902194253448, −7.987133507437764586259189477267, −7.38606334275503584840956522665, −5.66768322797412327138483915211, −4.01190780461266302182718900185, −2.92098896022043804587208625969,
2.63148429925184712151931523258, 3.87840837409952269174922803917, 5.20017227232130285701441001991, 7.08226816278506956060590122683, 8.180816343729966112239937434400, 9.273278854489555988499284415581, 10.57084099304183619846417218463, 11.77462005952393222689032052090, 12.54440739943567119127954549376, 13.67384755291122868592569579971
