| L(s) = 1 | + (−1.23 − 0.681i)2-s + (1.07 + 1.68i)4-s + 1.41·7-s + (−0.175 − 2.82i)8-s + 6.37i·11-s + 3.54·13-s + (−1.75 − 0.964i)14-s + (−1.70 + 3.61i)16-s − 3.92·17-s − 1.27·19-s + (4.34 − 7.89i)22-s + 6.28i·23-s + (−4.38 − 2.41i)26-s + (1.51 + 2.38i)28-s − 9.00·29-s + ⋯ |
| L(s) = 1 | + (−0.876 − 0.482i)2-s + (0.535 + 0.844i)4-s + 0.534·7-s + (−0.0619 − 0.998i)8-s + 1.92i·11-s + 0.982·13-s + (−0.468 − 0.257i)14-s + (−0.426 + 0.904i)16-s − 0.952·17-s − 0.292·19-s + (0.925 − 1.68i)22-s + 1.31i·23-s + (−0.860 − 0.473i)26-s + (0.286 + 0.451i)28-s − 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7534214739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7534214739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 0.681i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 6.37iT - 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 + 3.92T + 17T^{2} \) |
| 19 | \( 1 + 1.27T + 19T^{2} \) |
| 23 | \( 1 - 6.28iT - 23T^{2} \) |
| 29 | \( 1 + 9.00T + 29T^{2} \) |
| 31 | \( 1 - 3.92iT - 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 + 5.27iT - 41T^{2} \) |
| 43 | \( 1 + 1.55iT - 43T^{2} \) |
| 47 | \( 1 + 9.73iT - 47T^{2} \) |
| 53 | \( 1 - 5.55iT - 53T^{2} \) |
| 59 | \( 1 + 0.313iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 7.00iT - 67T^{2} \) |
| 71 | \( 1 - 0.990T + 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 8.18iT - 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 + 0.386iT - 89T^{2} \) |
| 97 | \( 1 - 10.4iT - 97T^{2} \) |
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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
−9.346748803139519361427894390755, −8.949638755936751527920440031686, −7.962354872095597201518143929394, −7.27938374441723512083898425447, −6.68570734308414488662176556574, −5.42834050101483112163669597859, −4.34055293553828998818000158017, −3.57673860221591193704247228351, −2.12745846065352322832573764860, −1.57692068828893992281186862604,
0.36933151790065105690600720294, 1.61922503158399402444805236734, 2.86066635028956530764447919539, 4.08276787238054171994724935002, 5.25178703518280400676654049217, 6.11939812413743269129951319852, 6.52789292440718160257559028331, 7.73509572404580120388780010074, 8.416407865408537247339389716727, 8.766914800546281924161099131859
