| L(s) = 1 | + (1.41 − 1.41i)2-s + (1.99 + 2.24i)3-s + (0.00911 − 3.99i)4-s + (−2.76 − 4.16i)5-s + (5.98 + 0.360i)6-s + (2.17 − 2.17i)7-s + (−5.63 − 5.67i)8-s + (−1.05 + 8.93i)9-s + (−9.80 − 1.98i)10-s + (5.18 − 15.9i)11-s + (8.98 − 7.95i)12-s + (5.69 + 2.89i)13-s + (0.00702 − 6.16i)14-s + (3.82 − 14.5i)15-s + (−15.9 − 0.0729i)16-s + (2.94 − 18.6i)17-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.706i)2-s + (0.664 + 0.747i)3-s + (0.00227 − 0.999i)4-s + (−0.553 − 0.832i)5-s + (0.998 + 0.0600i)6-s + (0.311 − 0.311i)7-s + (−0.704 − 0.709i)8-s + (−0.117 + 0.993i)9-s + (−0.980 − 0.198i)10-s + (0.471 − 1.45i)11-s + (0.749 − 0.662i)12-s + (0.437 + 0.223i)13-s + (0.000501 − 0.440i)14-s + (0.254 − 0.967i)15-s + (−0.999 − 0.00455i)16-s + (0.173 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0501 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0501 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.82934 - 1.92348i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82934 - 1.92348i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 1.41i)T \) |
| 3 | \( 1 + (-1.99 - 2.24i)T \) |
| 5 | \( 1 + (2.76 + 4.16i)T \) |
| good | 7 | \( 1 + (-2.17 + 2.17i)T - 49iT^{2} \) |
| 11 | \( 1 + (-5.18 + 15.9i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-5.69 - 2.89i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-2.94 + 18.6i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (0.956 - 0.695i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-3.62 - 7.11i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-28.2 - 20.5i)T + (259. + 799. i)T^{2} \) |
| 31 | \( 1 + (2.12 + 2.91i)T + (-296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-9.94 + 19.5i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-55.8 + 18.1i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (14.0 + 14.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-4.21 - 26.6i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-7.38 - 46.6i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (94.7 - 30.7i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (24.4 - 75.2i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-3.32 + 20.9i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-65.5 - 47.5i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-20.7 - 40.7i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (13.2 + 9.63i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-22.3 + 141. i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-35.3 + 108. i)T + (-6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-14.4 - 91.1i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−11.25198754175608611873895852434, −10.62923924298282682682691477177, −9.254931322941382048433502184395, −8.839528014410071767017614559119, −7.59014096896958508204494376389, −5.87142846044765388923536292658, −4.80015741268703819612259573033, −3.95341408672144494785066105778, −2.97829369060070593637482217186, −1.05039898704513347772711020974,
2.17341786564576182405292735641, 3.44214729339086498863295109260, 4.49425764219239477336539647963, 6.20244076835595446812107240598, 6.83166511864812928895214048471, 7.82935901665286113546163583178, 8.387925755314636486891747777966, 9.708253821006414043476017622278, 11.15288432363058145032359466803, 12.17752897645890674117764065883
