| L(s) = 1 | + (0.391 − 1.46i)2-s + (−1.50 − 0.852i)3-s + (−0.246 − 0.142i)4-s + (−1.82 − 1.29i)5-s + (−1.83 + 1.86i)6-s + (−1.17 − 2.36i)7-s + (1.83 − 1.83i)8-s + (1.54 + 2.57i)9-s + (−2.60 + 2.15i)10-s + (0.791 + 0.457i)11-s + (0.250 + 0.425i)12-s + (3.07 + 3.07i)13-s + (−3.92 + 0.791i)14-s + (1.64 + 3.50i)15-s + (−2.24 − 3.88i)16-s + (−1.16 + 0.311i)17-s + ⋯ |
| L(s) = 1 | + (0.276 − 1.03i)2-s + (−0.870 − 0.492i)3-s + (−0.123 − 0.0712i)4-s + (−0.815 − 0.578i)5-s + (−0.749 + 0.762i)6-s + (−0.444 − 0.895i)7-s + (0.648 − 0.648i)8-s + (0.514 + 0.857i)9-s + (−0.822 + 0.682i)10-s + (0.238 + 0.137i)11-s + (0.0723 + 0.122i)12-s + (0.854 + 0.854i)13-s + (−1.04 + 0.211i)14-s + (0.425 + 0.905i)15-s + (−0.561 − 0.971i)16-s + (−0.281 + 0.0755i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.346474 - 0.811701i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.346474 - 0.811701i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.50 + 0.852i)T \) |
| 5 | \( 1 + (1.82 + 1.29i)T \) |
| 7 | \( 1 + (1.17 + 2.36i)T \) |
| good | 2 | \( 1 + (-0.391 + 1.46i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (-0.791 - 0.457i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.07 - 3.07i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.16 - 0.311i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.95 + 3.43i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.88 - 0.505i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 + (2.31 - 4.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.774 + 0.207i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 0.922iT - 41T^{2} \) |
| 43 | \( 1 + (4.80 + 4.80i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.71 - 10.1i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.85 - 10.6i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.94 + 8.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.533 - 0.924i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.83 + 6.83i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 0.557iT - 71T^{2} \) |
| 73 | \( 1 + (-2.10 + 0.564i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.62 + 1.51i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.38 + 2.38i)T - 83iT^{2} \) |
| 89 | \( 1 + (-5.64 - 9.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.58 - 1.58i)T - 97iT^{2} \) |
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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
−13.10545829534800585648036722395, −12.17736753742248085076640534959, −11.38074132048281626785467259237, −10.77314203248172828646369853494, −9.346335248650964707659959324877, −7.55072265305664674787126006136, −6.73526595552753343504295297846, −4.78074203409352056305684315805, −3.63122185244531382401112235012, −1.20366283258916783520851124833,
3.57077880886964560767576589599, 5.29267953260960981858633095161, 6.12536123997161396342071493708, 7.17487121036648762662375216247, 8.460409350111660589183480168983, 10.02412918895013540773180640970, 11.18920240970942082775366610696, 11.84234174038943797960863573592, 13.19707113383234324256325354172, 14.72305063301499441201271662251
