| L(s) = 1 | + (−0.669 + 0.743i)2-s + (−3.06 + 1.36i)3-s + (−0.104 − 0.994i)4-s + (0.248 − 0.0528i)5-s + (1.03 − 3.18i)6-s + (−2.64 + 0.0451i)7-s + (0.809 + 0.587i)8-s + (5.50 − 6.11i)9-s + (−0.127 + 0.220i)10-s + (0.623 − 3.25i)11-s + (1.67 + 2.90i)12-s + (−0.246 − 0.758i)13-s + (1.73 − 1.99i)14-s + (−0.689 + 0.501i)15-s + (−0.978 + 0.207i)16-s + (−2.64 − 2.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.473 + 0.525i)2-s + (−1.76 + 0.786i)3-s + (−0.0522 − 0.497i)4-s + (0.111 − 0.0236i)5-s + (0.422 − 1.30i)6-s + (−0.999 + 0.0170i)7-s + (0.286 + 0.207i)8-s + (1.83 − 2.03i)9-s + (−0.0402 + 0.0696i)10-s + (0.187 − 0.982i)11-s + (0.483 + 0.837i)12-s + (−0.0683 − 0.210i)13-s + (0.464 − 0.533i)14-s + (−0.178 + 0.129i)15-s + (−0.244 + 0.0519i)16-s + (−0.640 − 0.711i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.133084 - 0.115900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.133084 - 0.115900i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 7 | \( 1 + (2.64 - 0.0451i)T \) |
| 11 | \( 1 + (-0.623 + 3.25i)T \) |
| good | 3 | \( 1 + (3.06 - 1.36i)T + (2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (-0.248 + 0.0528i)T + (4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (0.246 + 0.758i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.64 + 2.93i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.190 - 1.80i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (3.47 + 6.01i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.76 - 2.00i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.14 + 0.244i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (-0.300 - 0.133i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (6.58 + 4.78i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 + (-1.05 + 9.99i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-3.78 - 0.805i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.417 - 3.96i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (5.25 - 1.11i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.0454 - 0.0787i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.72 + 8.37i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.167 - 1.59i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (9.95 - 11.0i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (2.38 - 7.32i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.01 - 3.49i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.59 + 14.1i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.47363731688405475899330723615, −11.53563193150720541138380836228, −10.57860812010037817616650497393, −9.879285196285274871279893959884, −8.887304527565292622499989446395, −7.00433409981571430328426841908, −6.14142057978501879769270550351, −5.39768930778907228209055458423, −3.92205052366280623676100989617, −0.24039318068720190398995006982,
1.84115562443288317534463211752, 4.30383903105448719690115419660, 5.84709738430355502081982646322, 6.74777529736465679325972037016, 7.67415400588307906480432598799, 9.540879623098627157851680394010, 10.29595812759999289118597597173, 11.34423705834389612583862231572, 12.07513189288565215556466133939, 12.85827593635752692735731188045
