| L(s) = 1 | + (−0.156 + 0.987i)2-s + (−2.33 − 1.18i)3-s + (−0.951 − 0.309i)4-s + (1.55 − 1.61i)5-s + (1.53 − 2.11i)6-s + (−1.46 − 2.88i)7-s + (0.453 − 0.891i)8-s + (2.26 + 3.12i)9-s + (1.34 + 1.78i)10-s + (−0.0951 − 3.31i)11-s + (1.85 + 1.85i)12-s + (1.57 + 0.250i)13-s + (3.07 − 1.00i)14-s + (−5.53 + 1.91i)15-s + (0.809 + 0.587i)16-s + (−5.63 + 0.892i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 + 0.698i)2-s + (−1.34 − 0.686i)3-s + (−0.475 − 0.154i)4-s + (0.693 − 0.720i)5-s + (0.628 − 0.865i)6-s + (−0.555 − 1.08i)7-s + (0.160 − 0.315i)8-s + (0.756 + 1.04i)9-s + (0.426 + 0.563i)10-s + (−0.0286 − 0.999i)11-s + (0.534 + 0.534i)12-s + (0.438 + 0.0693i)13-s + (0.822 − 0.267i)14-s + (−1.42 + 0.494i)15-s + (0.202 + 0.146i)16-s + (−1.36 + 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.492931 - 0.342309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.492931 - 0.342309i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.156 - 0.987i)T \) |
| 5 | \( 1 + (-1.55 + 1.61i)T \) |
| 11 | \( 1 + (0.0951 + 3.31i)T \) |
| good | 3 | \( 1 + (2.33 + 1.18i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (1.46 + 2.88i)T + (-4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (-1.57 - 0.250i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (5.63 - 0.892i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.49 - 4.60i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.82 + 1.82i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.0395 + 0.121i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.90 + 2.11i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-9.82 + 5.00i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-3.82 + 1.24i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-7.57 - 7.57i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.13 - 4.18i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.147 + 0.932i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (6.38 + 2.07i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.16 - 1.60i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.46 - 6.46i)T + 67iT^{2} \) |
| 71 | \( 1 + (10.1 + 7.40i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.83 - 1.95i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-4.04 + 2.93i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.45 + 9.20i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + 8.27iT - 89T^{2} \) |
| 97 | \( 1 + (7.60 + 1.20i)T + (92.2 + 29.9i)T^{2} \) |
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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
−13.24662698341580262182928143722, −12.81437553441675490784975418550, −11.32497081291648685036164427455, −10.40962295537916472487921217434, −9.095296870392844033821421432138, −7.70178011070567428959580940234, −6.34462874514521583729405305520, −5.94266319181975685919798403632, −4.41668545936106959238018901333, −0.858222942570969430565190662341,
2.61133194814814804867170852619, 4.61061903084138904712086042044, 5.77386157107834626860944738955, 6.81092217407351476584943896479, 9.124921381367048202812244794646, 9.808070401983666609244876808764, 10.86849814414753637660997297043, 11.50883962747992266863999126696, 12.56932527453499010086878058374, 13.56236790143474748001287097317
