L(s) = 1 | + (2.17 − 0.582i)2-s + (−0.258 + 0.965i)3-s + (2.65 − 1.53i)4-s + (−1.96 + 1.06i)5-s + 2.25i·6-s + (−0.660 − 2.56i)7-s + (1.69 − 1.69i)8-s + (−0.866 − 0.499i)9-s + (−3.65 + 3.46i)10-s + (0.329 + 0.571i)11-s + (0.793 + 2.96i)12-s + (−2.55 − 2.55i)13-s + (−2.92 − 5.18i)14-s + (−0.521 − 2.17i)15-s + (−0.366 + 0.635i)16-s + (5.43 + 1.45i)17-s + ⋯ |
L(s) = 1 | + (1.53 − 0.411i)2-s + (−0.149 + 0.557i)3-s + (1.32 − 0.766i)4-s + (−0.878 + 0.476i)5-s + 0.918i·6-s + (−0.249 − 0.968i)7-s + (0.599 − 0.599i)8-s + (−0.288 − 0.166i)9-s + (−1.15 + 1.09i)10-s + (0.0994 + 0.172i)11-s + (0.229 + 0.854i)12-s + (−0.709 − 0.709i)13-s + (−0.782 − 1.38i)14-s + (−0.134 − 0.561i)15-s + (−0.0916 + 0.158i)16-s + (1.31 + 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.76867 - 0.117647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76867 - 0.117647i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 + (1.96 - 1.06i)T \) |
| 7 | \( 1 + (0.660 + 2.56i)T \) |
good | 2 | \( 1 + (-2.17 + 0.582i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (-0.329 - 0.571i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.55 + 2.55i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.43 - 1.45i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.48 - 2.56i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0726 - 0.271i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 5.03iT - 29T^{2} \) |
| 31 | \( 1 + (-6.53 + 3.77i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.79 - 2.08i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-8.53 + 8.53i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.11 + 11.6i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (4.65 + 1.24i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.782 + 1.35i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.02 + 0.589i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.762 + 2.84i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 3.13T + 71T^{2} \) |
| 73 | \( 1 + (-0.417 + 1.55i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.17 - 3.56i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.14 + 2.14i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.24 - 3.88i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.33 - 3.33i)T - 97iT^{2} \) |
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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
−13.85799545457472082153795562437, −12.56806475432652483129827337550, −11.93883736270146586348165339596, −10.74929342646399350407276151813, −10.11001396988900134471975056899, −7.977981826766933059088771040874, −6.68402786352950591831651550547, −5.23185889197297481782173655441, −4.02512451627713244513973638948, −3.19817204168387144604009034005,
2.92086999946600904324485627205, 4.50944920904112022641681840216, 5.58772814079095824525204304446, 6.76701214896687576890032590738, 7.895362420369029202300181598630, 9.294417061821116042473985976804, 11.39308823587977481431912873164, 12.25995808876083094424697328619, 12.51050755899044241059555797711, 13.83097081935708290536199593073