| L(s) = 1 | + (1.49 − 0.683i)2-s + (−0.900 − 3.06i)3-s + (0.465 − 0.536i)4-s + (1.43 + 1.71i)5-s + (−3.44 − 3.97i)6-s + (−0.632 + 0.0908i)7-s + (−0.598 + 2.03i)8-s + (−6.07 + 3.90i)9-s + (3.32 + 1.58i)10-s + (1.69 − 3.71i)11-s + (−2.06 − 0.943i)12-s + (3.19 + 0.459i)13-s + (−0.884 + 0.568i)14-s + (3.96 − 5.94i)15-s + (0.699 + 4.86i)16-s + (−0.965 + 0.836i)17-s + ⋯ |
| L(s) = 1 | + (1.05 − 0.483i)2-s + (−0.519 − 1.77i)3-s + (0.232 − 0.268i)4-s + (0.641 + 0.767i)5-s + (−1.40 − 1.62i)6-s + (−0.238 + 0.0343i)7-s + (−0.211 + 0.720i)8-s + (−2.02 + 1.30i)9-s + (1.05 + 0.501i)10-s + (0.512 − 1.12i)11-s + (−0.596 − 0.272i)12-s + (0.885 + 0.127i)13-s + (−0.236 + 0.151i)14-s + (1.02 − 1.53i)15-s + (0.174 + 1.21i)16-s + (−0.234 + 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0838 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0838 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09477 - 1.00652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09477 - 1.00652i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.43 - 1.71i)T \) |
| 23 | \( 1 + (3.19 - 3.57i)T \) |
| good | 2 | \( 1 + (-1.49 + 0.683i)T + (1.30 - 1.51i)T^{2} \) |
| 3 | \( 1 + (0.900 + 3.06i)T + (-2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (0.632 - 0.0908i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.69 + 3.71i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-3.19 - 0.459i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.965 - 0.836i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.307 + 0.354i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (2.61 + 3.02i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (4.96 + 1.45i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (5.16 + 8.04i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-7.36 - 4.73i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (0.206 + 0.704i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 0.589iT - 47T^{2} \) |
| 53 | \( 1 + (-6.64 + 0.955i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.734 - 5.11i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (2.44 + 0.717i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (2.18 - 0.998i)T + (43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (3.46 + 7.58i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (0.557 + 0.483i)T + (10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.33 + 16.2i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-5.64 - 8.77i)T + (-34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-5.75 + 1.68i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-3.48 + 5.42i)T + (-40.2 - 88.2i)T^{2} \) |
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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
−13.47698609288809238486705115128, −12.50309152544073238323023607724, −11.43605006750476187118589439528, −10.98971196248207638457219036722, −8.849520014118930962003122456851, −7.51948376383836797587644251201, −6.11693069505861444571756175536, −5.82911434866450744594950866161, −3.41394961671032283523545676736, −1.95478281622241575475720964661,
3.75750763722729589303142586985, 4.65164168069416948899648773096, 5.49734119698475228281973746017, 6.48940616273987586724344337943, 8.897905138750046496729282424742, 9.663492763173808920562775008028, 10.49454714487761792960343675288, 11.93358179497711137294945305446, 12.88764437705860586812583433768, 14.09281390862141807533276416947
