| L(s) = 1 | + (0.559 − 1.29i)2-s + (−0.154 + 1.72i)3-s + (−1.37 − 1.45i)4-s + (−0.778 + 2.09i)5-s + (2.15 + 1.16i)6-s + 3.14·7-s + (−2.65 + 0.971i)8-s + (−2.95 − 0.532i)9-s + (2.28 + 2.18i)10-s + (1.76 + 1.28i)11-s + (2.71 − 2.14i)12-s + (2.63 + 3.62i)13-s + (1.76 − 4.08i)14-s + (−3.49 − 1.66i)15-s + (−0.223 + 3.99i)16-s + (−1.48 + 4.57i)17-s + ⋯ |
| L(s) = 1 | + (0.395 − 0.918i)2-s + (−0.0891 + 0.996i)3-s + (−0.687 − 0.726i)4-s + (−0.348 + 0.937i)5-s + (0.879 + 0.475i)6-s + 1.19·7-s + (−0.939 + 0.343i)8-s + (−0.984 − 0.177i)9-s + (0.723 + 0.690i)10-s + (0.531 + 0.386i)11-s + (0.784 − 0.619i)12-s + (0.731 + 1.00i)13-s + (0.470 − 1.09i)14-s + (−0.902 − 0.430i)15-s + (−0.0559 + 0.998i)16-s + (−0.360 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40824 + 0.329473i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40824 + 0.329473i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.559 + 1.29i)T \) |
| 3 | \( 1 + (0.154 - 1.72i)T \) |
| 5 | \( 1 + (0.778 - 2.09i)T \) |
| good | 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 + (-1.76 - 1.28i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.63 - 3.62i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.48 - 4.57i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.98 - 0.968i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.21 + 5.80i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.17 - 0.382i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (5.04 + 1.63i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.18 + 7.12i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (3.03 + 4.18i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 0.651T + 43T^{2} \) |
| 47 | \( 1 + (-0.972 + 0.315i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.802 + 2.47i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.99 + 2.17i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 7.51i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.32 + 10.2i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.91 - 12.0i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.85 + 9.44i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (10.8 - 3.53i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.25 - 1.38i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.83 + 2.52i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.13 - 1.01i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.40449881673385628476977981033, −11.04609318018273666948271447034, −10.32735932581354705935287625632, −9.164939932948807475826323032080, −8.385066160598536537873529971274, −6.72404600395025936377976875398, −5.47065226963363995929792795957, −4.28407339361618805560529482886, −3.65861362624992196414264422131, −2.03047150620192908778672808234,
1.10373692937420435909475717659, 3.39383707464181139751232929993, 5.02388987284790255034689182393, 5.51111938869844057736662266171, 6.93519414366892008473184262774, 7.79916164800579600271303241681, 8.452990850998388453331015880798, 9.233254699554241325217296360178, 11.35404197447756146602893600408, 11.70577046952703780006796870486
