L(s) = 1 | + (−4.36 − 7.56i)5-s + (14.4 + 11.5i)7-s + (7.60 + 4.39i)11-s + 11.8i·13-s + (22.2 − 38.6i)17-s + (10.0 − 5.82i)19-s + (123. − 71.3i)23-s + (24.3 − 42.1i)25-s − 234. i·29-s + (252. + 145. i)31-s + (23.9 − 160. i)35-s + (44.4 + 76.9i)37-s + 145.·41-s + 144.·43-s + (−120. − 208. i)47-s + ⋯ |
L(s) = 1 | + (−0.390 − 0.676i)5-s + (0.782 + 0.622i)7-s + (0.208 + 0.120i)11-s + 0.252i·13-s + (0.318 − 0.550i)17-s + (0.121 − 0.0703i)19-s + (1.11 − 0.646i)23-s + (0.194 − 0.337i)25-s − 1.49i·29-s + (1.46 + 0.845i)31-s + (0.115 − 0.772i)35-s + (0.197 + 0.342i)37-s + 0.555·41-s + 0.512·43-s + (−0.372 − 0.646i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.867037217\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.867037217\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-14.4 - 11.5i)T \) |
good | 5 | \( 1 + (4.36 + 7.56i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-7.60 - 4.39i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 11.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-22.2 + 38.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-10.0 + 5.82i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-123. + 71.3i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 234. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-252. - 145. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-44.4 - 76.9i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (120. + 208. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-263. - 152. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-3.54 + 6.13i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (149. - 86.4i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-243. + 421. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 653. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (99.0 + 57.1i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (147. + 255. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 877.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-710. - 1.23e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 738. iT - 9.12e5T^{2} \) |
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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
−11.80021833727769543289217631636, −10.68198455182369293342268018342, −9.429364087759273932167984138656, −8.586434509977010470769449185899, −7.79292674278735029051100916855, −6.48059956849416865318556784470, −5.13830018265612590203636044525, −4.36870729287541225996474456809, −2.61533623480734461307672275936, −0.955814519366399086550285932077,
1.19641289905962759398958801183, 3.03571174662691110399937821537, 4.21252336694129147497075410536, 5.48701144324457212000685987475, 6.87573408523157358938316202506, 7.63335363689393266111771649316, 8.628674301001009542440299806272, 9.911260988702193425416323561403, 10.92478888420561270062172295100, 11.37405742193130628778538349263