| L(s) = 1 | + (−0.987 + 0.156i)2-s + (−0.176 − 0.346i)3-s + (0.951 − 0.309i)4-s + (0.0767 − 2.23i)5-s + (0.228 + 0.314i)6-s + (−0.0136 − 0.00697i)7-s + (−0.891 + 0.453i)8-s + (1.67 − 2.30i)9-s + (0.273 + 2.21i)10-s + (2.43 − 2.25i)11-s + (−0.274 − 0.274i)12-s + (0.366 + 2.31i)13-s + (0.0146 + 0.00474i)14-s + (−0.787 + 0.367i)15-s + (0.809 − 0.587i)16-s + (0.615 − 3.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 + 0.110i)2-s + (−0.101 − 0.199i)3-s + (0.475 − 0.154i)4-s + (0.0343 − 0.999i)5-s + (0.0932 + 0.128i)6-s + (−0.00517 − 0.00263i)7-s + (−0.315 + 0.160i)8-s + (0.558 − 0.768i)9-s + (0.0865 + 0.701i)10-s + (0.733 − 0.679i)11-s + (−0.0793 − 0.0793i)12-s + (0.101 + 0.641i)13-s + (0.00390 + 0.00126i)14-s + (−0.203 + 0.0949i)15-s + (0.202 − 0.146i)16-s + (0.149 − 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.707000 - 0.317971i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.707000 - 0.317971i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.987 - 0.156i)T \) |
| 5 | \( 1 + (-0.0767 + 2.23i)T \) |
| 11 | \( 1 + (-2.43 + 2.25i)T \) |
| good | 3 | \( 1 + (0.176 + 0.346i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (0.0136 + 0.00697i)T + (4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.366 - 2.31i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.615 + 3.88i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (1.78 - 5.49i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (4.75 - 4.75i)T - 23iT^{2} \) |
| 29 | \( 1 + (-1.28 - 3.94i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.70 - 3.41i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.512 + 1.00i)T + (-21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (4.46 + 1.45i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-5.68 - 5.68i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.37 + 0.703i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-13.8 + 2.19i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (1.86 - 0.606i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.84 - 6.66i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.67 - 3.67i)T + 67iT^{2} \) |
| 71 | \( 1 + (5.20 - 3.77i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.09 + 4.10i)T + (-42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (3.84 + 2.79i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (13.1 + 2.08i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 - 9.48iT - 89T^{2} \) |
| 97 | \( 1 + (-0.696 - 4.39i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−13.50947825315040293126500216252, −12.16317749445887214222456024783, −11.70341844676058818972265881067, −10.04274676531490840029916668432, −9.194794339573740260748421781386, −8.268276748983136590346177439884, −6.90718709623447829774128356799, −5.73019614549688581421628790217, −3.94687721566941772639432061320, −1.32678809804840175430578132392,
2.33247816287611911136931004971, 4.21462900478563658545352072343, 6.19309470777183387510697281519, 7.25094568214617244623950633754, 8.355242092818255602358841886524, 9.875638584001923501673862974348, 10.45655323351507765926219062678, 11.42337255914033301884705662977, 12.63885436917099311419421119770, 13.87052697573410268801640343463
