| L(s) = 1 | + (−0.417 + 1.82i)2-s + (−8.97 + 11.2i)3-s + (25.6 + 12.3i)4-s + (41.8 − 52.4i)5-s + (−16.8 − 21.1i)6-s + (124. − 37.1i)7-s + (−70.7 + 88.6i)8-s + (7.96 + 34.9i)9-s + (78.4 + 98.3i)10-s + (−23.8 + 104. i)11-s + (−369. + 177. i)12-s + (−166. + 730. i)13-s + (16.1 + 242. i)14-s + (214. + 941. i)15-s + (435. + 546. i)16-s + (1.52e3 − 732. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0737 + 0.323i)2-s + (−0.575 + 0.721i)3-s + (0.801 + 0.386i)4-s + (0.748 − 0.938i)5-s + (−0.190 − 0.239i)6-s + (0.958 − 0.286i)7-s + (−0.390 + 0.489i)8-s + (0.0327 + 0.143i)9-s + (0.248 + 0.311i)10-s + (−0.0593 + 0.260i)11-s + (−0.740 + 0.356i)12-s + (−0.273 + 1.19i)13-s + (0.0219 + 0.330i)14-s + (0.246 + 1.08i)15-s + (0.425 + 0.533i)16-s + (1.27 − 0.614i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.51813 + 1.13824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51813 + 1.13824i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-124. + 37.1i)T \) |
| good | 2 | \( 1 + (0.417 - 1.82i)T + (-28.8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (8.97 - 11.2i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (-41.8 + 52.4i)T + (-695. - 3.04e3i)T^{2} \) |
| 11 | \( 1 + (23.8 - 104. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (166. - 730. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-1.52e3 + 732. i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 + 1.96e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.37e3 - 660. i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 29 | \( 1 + (2.35e3 - 1.13e3i)T + (1.27e7 - 1.60e7i)T^{2} \) |
| 31 | \( 1 - 7.64e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (1.60e3 - 772. i)T + (4.32e7 - 5.42e7i)T^{2} \) |
| 41 | \( 1 + (-1.11e4 + 1.39e4i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 43 | \( 1 + (1.22e4 + 1.53e4i)T + (-3.27e7 + 1.43e8i)T^{2} \) |
| 47 | \( 1 + (1.13e3 - 4.99e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-1.11e4 - 5.35e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (6.60e3 + 8.28e3i)T + (-1.59e8 + 6.96e8i)T^{2} \) |
| 61 | \( 1 + (-1.29e4 + 6.22e3i)T + (5.26e8 - 6.60e8i)T^{2} \) |
| 67 | \( 1 + 6.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-3.01e3 - 1.45e3i)T + (1.12e9 + 1.41e9i)T^{2} \) |
| 73 | \( 1 + (2.20e3 + 9.67e3i)T + (-1.86e9 + 8.99e8i)T^{2} \) |
| 79 | \( 1 + 4.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (1.77e4 + 7.78e4i)T + (-3.54e9 + 1.70e9i)T^{2} \) |
| 89 | \( 1 + (1.12e4 + 4.91e4i)T + (-5.03e9 + 2.42e9i)T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−15.04590754171133195666314506437, −13.81205339712414797672868857297, −12.28899053017910525004791755003, −11.31597877762573104549218876180, −10.15793460663065222014900937518, −8.739275028975298430997204167641, −7.31535289226441608349523906097, −5.62315242467239136678567026239, −4.54124309372062324755913005109, −1.82852412071712532712155745817,
1.23445284211281860977564917765, 2.71171568804931260105895284228, 5.69955479114900127066262765173, 6.50358212796520818914598273619, 7.898876286743224241175935002911, 10.05016396965303806136262933462, 10.86488001618859208794900407255, 11.89403834616149010193595895332, 12.95694424292282473815431356627, 14.65951560921112968035770184538
