L(s) = 1 | + (−0.804 − 2.47i)2-s + (0.229 − 0.166i)3-s + (−3.86 + 2.80i)4-s − 1.16·5-s + (−0.596 − 0.433i)6-s + (−0.525 − 1.61i)7-s + (5.85 + 4.25i)8-s + (−0.902 + 2.77i)9-s + (0.941 + 2.89i)10-s + (2.26 + 2.42i)11-s + (−0.418 + 1.28i)12-s + (1.00 + 3.08i)13-s + (−3.58 + 2.60i)14-s + (−0.267 + 0.194i)15-s + (2.86 − 8.81i)16-s + (−6.43 + 4.67i)17-s + ⋯ |
L(s) = 1 | + (−0.568 − 1.75i)2-s + (0.132 − 0.0960i)3-s + (−1.93 + 1.40i)4-s − 0.523·5-s + (−0.243 − 0.176i)6-s + (−0.198 − 0.611i)7-s + (2.06 + 1.50i)8-s + (−0.300 + 0.925i)9-s + (0.297 + 0.915i)10-s + (0.683 + 0.730i)11-s + (−0.120 + 0.371i)12-s + (0.278 + 0.856i)13-s + (−0.957 + 0.695i)14-s + (−0.0691 + 0.0502i)15-s + (0.716 − 2.20i)16-s + (−1.56 + 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.572696 - 0.0505721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.572696 - 0.0505721i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-2.26 - 2.42i)T \) |
| 41 | \( 1 + (0.000300 - 6.40i)T \) |
good | 2 | \( 1 + (0.804 + 2.47i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.229 + 0.166i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 7 | \( 1 + (0.525 + 1.61i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-1.00 - 3.08i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (6.43 - 4.67i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + 0.642T + 19T^{2} \) |
| 23 | \( 1 + (-7.42 + 5.39i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (5.26 - 3.82i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 + (-7.97 + 5.79i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (0.459 - 0.334i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (3.41 - 10.4i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (5.91 + 4.29i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + 7.36T + 59T^{2} \) |
| 61 | \( 1 + (3.18 - 9.79i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (3.14 - 9.68i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (5.12 - 3.72i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-7.38 - 5.36i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (1.33 - 4.10i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.18 - 6.73i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (7.43 + 5.40i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-12.1 - 8.85i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.10937054385435889714200817005, −10.44674226948582585642404968034, −9.338049125776097903579002572802, −8.722258946903771872025582516668, −7.78204824782912598714782237818, −6.64126431765513692408287835973, −4.47781602661990780621196313215, −4.10981967927298364359376740603, −2.64833933969607813221730672060, −1.53359827781495472349603505729,
0.45322365550991947372457673788, 3.33103542953429442528319784598, 4.68792977433084188435284996972, 5.84115666347562426193124385563, 6.46315037171752346071658483294, 7.43667367740876214943877822056, 8.364886277114328134981884410570, 9.165791779793964740976093203340, 9.470528476306157856232335381871, 11.06846925119390634984008462248