| L(s) = 1 | + (0.0512 + 0.487i)2-s + (−1.72 + 0.179i)3-s + (1.72 − 0.365i)4-s + (−0.178 + 1.69i)5-s + (−0.176 − 0.831i)6-s + (2.28 + 2.53i)7-s + (0.569 + 1.75i)8-s + (2.93 − 0.619i)9-s − 0.837·10-s + (−2.17 − 2.50i)11-s + (−2.89 + 0.939i)12-s + (−3.65 − 1.62i)13-s + (−1.11 + 1.24i)14-s + (0.00212 − 2.95i)15-s + (2.38 − 1.06i)16-s + (2.67 + 1.94i)17-s + ⋯ |
| L(s) = 1 | + (0.0362 + 0.344i)2-s + (−0.994 + 0.103i)3-s + (0.860 − 0.182i)4-s + (−0.0798 + 0.759i)5-s + (−0.0718 − 0.339i)6-s + (0.862 + 0.957i)7-s + (0.201 + 0.620i)8-s + (0.978 − 0.206i)9-s − 0.264·10-s + (−0.656 − 0.754i)11-s + (−0.836 + 0.271i)12-s + (−1.01 − 0.450i)13-s + (−0.299 + 0.332i)14-s + (0.000549 − 0.763i)15-s + (0.597 − 0.265i)16-s + (0.648 + 0.471i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.850893 + 0.429707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.850893 + 0.429707i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 - 0.179i)T \) |
| 11 | \( 1 + (2.17 + 2.50i)T \) |
| good | 2 | \( 1 + (-0.0512 - 0.487i)T + (-1.95 + 0.415i)T^{2} \) |
| 5 | \( 1 + (0.178 - 1.69i)T + (-4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (-2.28 - 2.53i)T + (-0.731 + 6.96i)T^{2} \) |
| 13 | \( 1 + (3.65 + 1.62i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-2.67 - 1.94i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.36 + 4.20i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.74 + 6.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.13 - 1.25i)T + (-3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (3.56 + 1.58i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.947 + 2.91i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.261 + 0.290i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (4.80 - 8.32i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.05 - 0.224i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-10.1 + 7.38i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.57 + 0.760i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (3.52 - 1.56i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (1.55 + 2.68i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.67 + 4.12i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.64 - 14.3i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.418 - 3.97i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (0.114 - 0.0508i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + 7.93T + 89T^{2} \) |
| 97 | \( 1 + (0.0358 + 0.341i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−14.60520531812837158177073313510, −12.71121058697910661207348065937, −11.73387333874654409045447574740, −10.96405814655826874307878035228, −10.22538696373022349993320658508, −8.257671203084174473117221740360, −7.10513294404353288482816964826, −5.96320068301005053662319326980, −5.07400859276563567572197796147, −2.53115022093598841947991719817,
1.63062985388091965032127731377, 4.26563953147160158407767814772, 5.41951233579235980458314550871, 7.15511347625494328900985745054, 7.75500169106717318208318355344, 9.928367913387592839146702269744, 10.62575058238373454742355895366, 11.89294117835913385613318793844, 12.22549000078633156538848784027, 13.44939103145816397934650921687
