| L(s) = 1 | + (4.38 − 0.872i)3-s + (−0.381 + 0.571i)5-s + (4.41 + 6.61i)7-s + (10.1 − 4.21i)9-s + (1.23 − 6.22i)11-s + (−2.83 + 2.83i)13-s + (−1.17 + 2.84i)15-s + (4.11 − 16.4i)17-s + (−27.5 − 11.3i)19-s + (25.1 + 25.1i)21-s + (−8.11 − 1.61i)23-s + (9.38 + 22.6i)25-s + (7.51 − 5.02i)27-s + (32.0 + 21.3i)29-s + (−5.66 − 28.4i)31-s + ⋯ |
| L(s) = 1 | + (1.46 − 0.290i)3-s + (−0.0763 + 0.114i)5-s + (0.631 + 0.944i)7-s + (1.13 − 0.468i)9-s + (0.112 − 0.566i)11-s + (−0.218 + 0.218i)13-s + (−0.0784 + 0.189i)15-s + (0.241 − 0.970i)17-s + (−1.44 − 0.599i)19-s + (1.19 + 1.19i)21-s + (−0.352 − 0.0701i)23-s + (0.375 + 0.906i)25-s + (0.278 − 0.185i)27-s + (1.10 + 0.737i)29-s + (−0.182 − 0.918i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0378i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.20202 - 0.0416839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20202 - 0.0416839i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-4.11 + 16.4i)T \) |
| good | 3 | \( 1 + (-4.38 + 0.872i)T + (8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (0.381 - 0.571i)T + (-9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (-4.41 - 6.61i)T + (-18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-1.23 + 6.22i)T + (-111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (2.83 - 2.83i)T - 169iT^{2} \) |
| 19 | \( 1 + (27.5 + 11.3i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (8.11 + 1.61i)T + (488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-32.0 - 21.3i)T + (321. + 776. i)T^{2} \) |
| 31 | \( 1 + (5.66 + 28.4i)T + (-887. + 367. i)T^{2} \) |
| 37 | \( 1 + (21.7 - 4.33i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (14.7 + 22.0i)T + (-643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (59.4 - 24.6i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (49.0 - 49.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (10.3 + 4.28i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-2.40 - 5.80i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (21.7 - 14.5i)T + (1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 47.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-131. + 26.1i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-58.4 + 87.4i)T + (-2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (22.7 - 114. i)T + (-5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-51.5 + 124. i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-97.2 - 97.2i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-140. - 93.8i)T + (3.60e3 + 8.69e3i)T^{2} \) |
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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
−13.17779898114406782736719603804, −12.07410704627123281259113664391, −10.99820059726668546211512388418, −9.450808176732556136587747402099, −8.683999807281033895654539140774, −7.951336730351271153689819789307, −6.64542161808518483617147016894, −4.93948911336888000772644383886, −3.22537631856840325949497069376, −2.07270211952659904123061078696,
1.95064106871897815658469386548, 3.65917909336286090567493747959, 4.61653713039220472428409654476, 6.70805545381966935417815120111, 8.097734817745682695471798169104, 8.441961246752816637749667172454, 9.968164556630564328489568419072, 10.53251226380680530375067111582, 12.16618045780656895154373001105, 13.20453152375007100896511545922
