| L(s) = 1 | + (−0.994 − 0.104i)2-s + (−1.46 − 1.31i)3-s + (0.978 + 0.207i)4-s + (2.21 − 0.305i)5-s + (1.31 + 1.46i)6-s + 2.84i·7-s + (−0.951 − 0.309i)8-s + (0.0930 + 0.885i)9-s + (−2.23 + 0.0725i)10-s + (−2.16 − 1.57i)11-s + (−1.15 − 1.59i)12-s + (2.37 − 0.249i)13-s + (0.297 − 2.83i)14-s + (−3.65 − 2.47i)15-s + (0.913 + 0.406i)16-s + (−0.0970 − 0.456i)17-s + ⋯ |
| L(s) = 1 | + (−0.703 − 0.0739i)2-s + (−0.846 − 0.761i)3-s + (0.489 + 0.103i)4-s + (0.990 − 0.136i)5-s + (0.538 + 0.598i)6-s + 1.07i·7-s + (−0.336 − 0.109i)8-s + (0.0310 + 0.295i)9-s + (−0.706 + 0.0229i)10-s + (−0.652 − 0.473i)11-s + (−0.334 − 0.460i)12-s + (0.657 − 0.0691i)13-s + (0.0795 − 0.757i)14-s + (−0.942 − 0.639i)15-s + (0.228 + 0.101i)16-s + (−0.0235 − 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.957318 - 0.338385i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.957318 - 0.338385i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.994 + 0.104i)T \) |
| 5 | \( 1 + (-2.21 + 0.305i)T \) |
| 19 | \( 1 + (-4.34 - 0.399i)T \) |
| good | 3 | \( 1 + (1.46 + 1.31i)T + (0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 - 2.84iT - 7T^{2} \) |
| 11 | \( 1 + (2.16 + 1.57i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.37 + 0.249i)T + (12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (0.0970 + 0.456i)T + (-15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (-2.48 - 5.57i)T + (-15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (4.95 + 1.05i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-2.09 + 6.44i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.32 - 3.19i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-8.31 - 3.70i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-2.34 + 1.35i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.63 - 12.3i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-2.47 + 11.6i)T + (-48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (6.81 + 3.03i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-9.34 + 4.16i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-7.30 + 6.57i)T + (7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-7.32 + 8.13i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-3.16 - 0.332i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (0.280 - 0.311i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (10.7 + 3.49i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.8 + 5.26i)T + (59.5 - 66.1i)T^{2} \) |
| 97 | \( 1 + (2.32 + 2.09i)T + (10.1 + 96.4i)T^{2} \) |
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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
−9.576048604553060557703954169476, −9.431779972159846792318634053661, −8.264893462800667940313528195878, −7.47036436953386464391328108398, −6.32500825171418764641188330466, −5.83779578350606248104473155927, −5.22767040892994900944633640059, −3.17771408116793732546429618125, −2.04955369634574679234530240275, −0.920000074256164268593405790335,
0.979102248380653275089330561509, 2.49944605468357496912623323755, 3.97744148901189088682160766843, 5.08089693566654466286721157593, 5.75440876763651383971087993009, 6.78902162153467577640567737486, 7.47885907566578077122297001082, 8.661646690023101623289440831359, 9.573139196947971667162184790423, 10.24818191092345429034934946981
