L(s) = 1 | + (8.5 + 16.4i)7-s − 19·13-s + (−53.5 + 92.6i)19-s + (62.5 + 108. i)25-s + (144.5 + 250. i)31-s + (−161.5 + 279. i)37-s + 71·43-s + (−198.5 + 279. i)49-s + (−91 + 157. i)61-s + (63.5 + 109. i)67-s + (135.5 + 234. i)73-s + (693.5 − 1.20e3i)79-s + (−161.5 − 312. i)91-s − 1.33e3·97-s + (900.5 − 1.55e3i)103-s + ⋯ |
L(s) = 1 | + (0.458 + 0.888i)7-s − 0.405·13-s + (−0.645 + 1.11i)19-s + (0.5 + 0.866i)25-s + (0.837 + 1.45i)31-s + (−0.717 + 1.24i)37-s + 0.251·43-s + (−0.578 + 0.815i)49-s + (−0.191 + 0.330i)61-s + (0.115 + 0.200i)67-s + (0.217 + 0.376i)73-s + (0.987 − 1.71i)79-s + (−0.186 − 0.360i)91-s − 1.39·97-s + (0.861 − 1.49i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.483523752\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483523752\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-8.5 - 16.4i)T \) |
good | 5 | \( 1 + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 19T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (53.5 - 92.6i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + (-144.5 - 250. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (161.5 - 279. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 71T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (91 - 157. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-63.5 - 109. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + (-135.5 - 234. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-693.5 + 1.20e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−11.99142862389887593208134056956, −10.90418008308671326571106410172, −9.949373293350446087258603932355, −8.827910867186952998273668740303, −8.089563139334207279806188194282, −6.82138810938263262783923667325, −5.66614437925952831607898670657, −4.66329370800323904244372253302, −3.09199133555047477090152710617, −1.65308511388879026424102448556,
0.59445987539238523480168153030, 2.37165980107151361586766339429, 4.02240455283875516625804132796, 4.96848099143220736352785486481, 6.43273735845060132822148579398, 7.39244879268777096048117576275, 8.358715037796500356309811049923, 9.487188821549907607488293309340, 10.52499942711777554767940032264, 11.22188209352479935401790272884