| L(s) = 1 | + (2.14 + 0.456i)2-s + (−1.28 − 1.15i)3-s + (2.57 + 1.14i)4-s + (0.900 + 2.04i)5-s + (−2.23 − 3.07i)6-s + (1.61 − 2.79i)7-s + (1.45 + 1.05i)8-s + (0.310 + 2.98i)9-s + (0.998 + 4.80i)10-s + (5.10 + 1.08i)11-s + (−1.98 − 4.46i)12-s + (−3.76 + 0.800i)13-s + (4.73 − 5.26i)14-s + (1.21 − 3.67i)15-s + (−1.12 − 1.25i)16-s + (−3.55 − 2.58i)17-s + ⋯ |
| L(s) = 1 | + (1.51 + 0.322i)2-s + (−0.742 − 0.669i)3-s + (1.28 + 0.573i)4-s + (0.402 + 0.915i)5-s + (−0.911 − 1.25i)6-s + (0.609 − 1.05i)7-s + (0.515 + 0.374i)8-s + (0.103 + 0.994i)9-s + (0.315 + 1.52i)10-s + (1.54 + 0.327i)11-s + (−0.572 − 1.28i)12-s + (−1.04 + 0.222i)13-s + (1.26 − 1.40i)14-s + (0.313 − 0.949i)15-s + (−0.281 − 0.312i)16-s + (−0.863 − 0.627i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.27388 + 0.0994159i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.27388 + 0.0994159i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 + 1.15i)T \) |
| 5 | \( 1 + (-0.900 - 2.04i)T \) |
| good | 2 | \( 1 + (-2.14 - 0.456i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (-1.61 + 2.79i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.10 - 1.08i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (3.76 - 0.800i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (3.55 + 2.58i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.66 + 1.93i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (5.59 - 6.21i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.104 + 0.997i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.352 - 3.35i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.878 - 2.70i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-5.99 + 1.27i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-2.26 + 3.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.111 + 1.05i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-6.03 + 4.38i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.06 - 0.652i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (8.43 + 1.79i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.364 - 3.46i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-11.0 + 8.01i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.320 + 0.986i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.840 - 7.99i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-5.24 + 2.33i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.92 - 9.00i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.199 - 1.90i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.25408461003072792383984251454, −11.61533815238377081018325997488, −10.84503343967159606331470786142, −9.562175590367867301888169106335, −7.42286980099417290376281710445, −6.94015997193926068006829988632, −6.16119148341291015256945111408, −4.88445708197733768837331502126, −3.95064937085860692157698545276, −2.10541673535744070198756469965,
2.13554379682787695366777373556, 4.10941849847059597579346532024, 4.66901947058869802847841306267, 5.81578307644629589156850720284, 6.28426023296866765330037878618, 8.567545949350684862232745733782, 9.366238380431492626746705821548, 10.70040242064838970247243886305, 11.74865725706083012199306750631, 12.22287296434191793132424757399
