| L(s) = 1 | + (−0.0493 − 0.0767i)2-s + (1.73 − 0.0750i)3-s + (0.827 − 1.81i)4-s + (−2.86 + 0.842i)5-s + (−0.0910 − 0.129i)6-s + (−1.75 + 1.52i)7-s + (−0.360 + 0.0518i)8-s + (2.98 − 0.259i)9-s + (0.206 + 0.178i)10-s + (1.83 + 1.17i)11-s + (1.29 − 3.19i)12-s + (−2.74 + 3.17i)13-s + (0.203 + 0.0597i)14-s + (−4.90 + 1.67i)15-s + (−2.58 − 2.98i)16-s + (−1.67 − 3.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.0348 − 0.0542i)2-s + (0.999 − 0.0433i)3-s + (0.413 − 0.905i)4-s + (−1.28 + 0.376i)5-s + (−0.0371 − 0.0526i)6-s + (−0.664 + 0.575i)7-s + (−0.127 + 0.0183i)8-s + (0.996 − 0.0866i)9-s + (0.0651 + 0.0564i)10-s + (0.553 + 0.355i)11-s + (0.374 − 0.922i)12-s + (−0.762 + 0.880i)13-s + (0.0543 + 0.0159i)14-s + (−1.26 + 0.431i)15-s + (−0.646 − 0.746i)16-s + (−0.406 − 0.890i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02573 - 0.147616i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02573 - 0.147616i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.73 + 0.0750i)T \) |
| 23 | \( 1 + (3.72 + 3.01i)T \) |
| good | 2 | \( 1 + (0.0493 + 0.0767i)T + (-0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (2.86 - 0.842i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (1.75 - 1.52i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.83 - 1.17i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.74 - 3.17i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.67 + 3.67i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.40 - 1.09i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-5.44 + 2.48i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.113 + 0.792i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-2.32 + 7.90i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.47 - 8.41i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (2.82 + 0.406i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 1.51iT - 47T^{2} \) |
| 53 | \( 1 + (-8.87 - 10.2i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (8.68 + 7.52i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-7.58 + 1.09i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.16 + 4.92i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-0.646 - 1.00i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (4.99 - 10.9i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.43 + 2.97i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (4.90 + 1.44i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.192 - 1.34i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.909 + 3.09i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−14.73150276319520019781486623325, −14.02495920500973274218440434284, −12.30597227856116228221595485411, −11.52160442825714429065268358827, −9.958643769278179669043487431636, −9.119762426019191224008574248079, −7.54896246197364442423390881108, −6.57928007296435406623530587859, −4.34710809668496946064658494217, −2.60983179780508099024426399387,
3.20904914722082033611310364394, 4.13028461918657010500756062246, 6.94886652824453982254396093064, 7.87028131810683156339965714174, 8.680178603551464329483155231617, 10.21260403107922845085901483721, 11.76203903212129189912482949546, 12.62176139725407643062976254794, 13.57744604471462276816852132604, 15.05586777401204799354698177960
