L(s) = 1 | + (−1.92 − 1.92i)2-s − 3-s + 5.42i·4-s + (−1.59 + 1.59i)5-s + (1.92 + 1.92i)6-s + (−1.66 + 1.66i)7-s + (6.60 − 6.60i)8-s + 9-s + 6.16·10-s + (2.43 − 2.25i)11-s − 5.42i·12-s + (1.55 + 3.25i)13-s + 6.42·14-s + (1.59 − 1.59i)15-s − 14.6·16-s − 4.73·17-s + ⋯ |
L(s) = 1 | + (−1.36 − 1.36i)2-s − 0.577·3-s + 2.71i·4-s + (−0.714 + 0.714i)5-s + (0.786 + 0.786i)6-s + (−0.630 + 0.630i)7-s + (2.33 − 2.33i)8-s + 0.333·9-s + 1.94·10-s + (0.733 − 0.680i)11-s − 1.56i·12-s + (0.431 + 0.901i)13-s + 1.71·14-s + (0.412 − 0.412i)15-s − 3.65·16-s − 1.14·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00457382 + 0.111071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00457382 + 0.111071i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + (-2.43 + 2.25i)T \) |
| 13 | \( 1 + (-1.55 - 3.25i)T \) |
good | 2 | \( 1 + (1.92 + 1.92i)T + 2iT^{2} \) |
| 5 | \( 1 + (1.59 - 1.59i)T - 5iT^{2} \) |
| 7 | \( 1 + (1.66 - 1.66i)T - 7iT^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 + (3.03 + 3.03i)T + 19iT^{2} \) |
| 23 | \( 1 - 1.62iT - 23T^{2} \) |
| 29 | \( 1 + 3.51iT - 29T^{2} \) |
| 31 | \( 1 + (-4.40 + 4.40i)T - 31iT^{2} \) |
| 37 | \( 1 + (3.16 + 3.16i)T + 37iT^{2} \) |
| 41 | \( 1 + (4.68 + 4.68i)T + 41iT^{2} \) |
| 43 | \( 1 + 4.69T + 43T^{2} \) |
| 47 | \( 1 + (6.86 + 6.86i)T + 47iT^{2} \) |
| 53 | \( 1 + 0.475T + 53T^{2} \) |
| 59 | \( 1 + (-8.18 - 8.18i)T + 59iT^{2} \) |
| 61 | \( 1 + 9.65iT - 61T^{2} \) |
| 67 | \( 1 + (-5.61 + 5.61i)T - 67iT^{2} \) |
| 71 | \( 1 + (-5.96 + 5.96i)T - 71iT^{2} \) |
| 73 | \( 1 + (1.78 - 1.78i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.07iT - 79T^{2} \) |
| 83 | \( 1 + (10.3 + 10.3i)T + 83iT^{2} \) |
| 89 | \( 1 + (13.0 + 13.0i)T + 89iT^{2} \) |
| 97 | \( 1 + (-8.39 + 8.39i)T - 97iT^{2} \) |
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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
−10.92428480365270874897108246299, −9.885148193172902771465986400060, −9.019254280389355710686815098154, −8.417330967439417414674751811699, −7.06846304283847811517506151596, −6.43397949219690468305893153062, −4.21419616862869527540520938696, −3.33272058884851443195164568910, −2.05987407328431976015132522923, −0.13432560182281011238523785844,
1.23205317726776421776007713602, 4.22590243862982196836219169895, 5.20959387174715940453889491219, 6.58831684917931805832981719812, 6.78898167185258925082792999486, 8.145408582560558677037619663277, 8.571080042442405806963058260822, 9.752576426420972435287752507444, 10.34277497732713178069643208500, 11.25404743962142447997676809940