L(s) = 1 | + (−0.448 − 0.258i)5-s + (−2.36 + 1.36i)7-s + (0.189 + 0.328i)11-s + (1.23 − 2.13i)13-s + 3.34i·17-s + 4.73i·19-s + (0.707 − 1.22i)23-s + (−2.36 − 4.09i)25-s + (5.91 − 3.41i)29-s + (0.464 + 0.267i)31-s + 1.41·35-s − 4.26·37-s + (−4.57 − 2.63i)41-s + (−4.56 + 2.63i)43-s + (−4.76 − 8.24i)47-s + ⋯ |
L(s) = 1 | + (−0.200 − 0.115i)5-s + (−0.894 + 0.516i)7-s + (0.0571 + 0.0989i)11-s + (0.341 − 0.591i)13-s + 0.811i·17-s + 1.08i·19-s + (0.147 − 0.255i)23-s + (−0.473 − 0.819i)25-s + (1.09 − 0.634i)29-s + (0.0833 + 0.0481i)31-s + 0.239·35-s − 0.701·37-s + (−0.713 − 0.412i)41-s + (−0.695 + 0.401i)43-s + (−0.694 − 1.20i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.448 + 0.258i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.189 - 0.328i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.23 + 2.13i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.34iT - 17T^{2} \) |
| 19 | \( 1 - 4.73iT - 19T^{2} \) |
| 23 | \( 1 + (-0.707 + 1.22i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.91 + 3.41i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.464 - 0.267i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 + (4.57 + 2.63i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.56 - 2.63i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.76 + 8.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.44iT - 53T^{2} \) |
| 59 | \( 1 + (6.83 - 11.8i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.59 + 7.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.63 + 5.56i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + (12.6 - 7.29i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.378 - 0.656i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.20iT - 89T^{2} \) |
| 97 | \( 1 + (7.19 + 12.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.362906586336521437373788387042, −8.041192034306181627666722839961, −6.83682195622763830079705282974, −6.18711485553965888326947066359, −5.57916960057542717760013777415, −4.46757783265318594820562574417, −3.59293539986804265583435024148, −2.80624169914994985926362246063, −1.58018835625577844324875652948, 0,
1.40427399586356978198974626526, 2.86694829903009245796667340569, 3.47510122113337864041750065073, 4.49702312940703289499796225090, 5.26887003406439530005000222199, 6.45136117011350249390387440287, 6.84794079155724188482637002545, 7.57898519499869683784182604006, 8.590902733351036139019356350768