L(s) = 1 | − 5.02i·2-s − 17.2·4-s + (5.75 − 9.58i)5-s − 5.38i·7-s + 46.4i·8-s + (−48.1 − 28.9i)10-s − 39.1·11-s − 86.6i·13-s − 27.0·14-s + 95.2·16-s + 15.4i·17-s + 26.8·19-s + (−99.2 + 165. i)20-s + 196. i·22-s + 111. i·23-s + ⋯ |
L(s) = 1 | − 1.77i·2-s − 2.15·4-s + (0.514 − 0.857i)5-s − 0.290i·7-s + 2.05i·8-s + (−1.52 − 0.914i)10-s − 1.07·11-s − 1.84i·13-s − 0.516·14-s + 1.48·16-s + 0.219i·17-s + 0.324·19-s + (−1.10 + 1.84i)20-s + 1.90i·22-s + 1.00i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7498886702\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7498886702\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-5.75 + 9.58i)T \) |
good | 2 | \( 1 + 5.02iT - 8T^{2} \) |
| 7 | \( 1 + 5.38iT - 343T^{2} \) |
| 11 | \( 1 + 39.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 86.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 15.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 26.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 49.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 293. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 27.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 60.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 96.2iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 251. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 76.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 238. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 640.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 769. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 662.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 995.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 814. iT - 9.12e5T^{2} \) |
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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
−10.22771345100632476276197958922, −9.576988192832384798117961868470, −8.530237586239383812443403269653, −7.70372346844128046540809593287, −5.59750488990947118901312662809, −5.01727188008601286386319104538, −3.66791461880248161969063831065, −2.64925779315085412040338174255, −1.36523031181053002959664472489, −0.25719845132464015890693293509,
2.33080982728629619186206370344, 4.07934119383377232829384990218, 5.25491269008813661179134616521, 6.05398777083983815989199863321, 6.97091591037464652460770483757, 7.50576339320867024645822128805, 8.759213770184743931311719689229, 9.379635042642571297164277783922, 10.44132743244250180183494851289, 11.52305966745059981382894212167