| L(s) = 1 | + (−1.31 − 0.517i)2-s + (0.717 − 2.17i)3-s + (1.46 + 1.36i)4-s + (0.159 − 0.413i)5-s + (−2.06 + 2.48i)6-s + (−0.0873 + 0.145i)7-s + (−1.22 − 2.55i)8-s + (−1.79 − 1.33i)9-s + (−0.424 + 0.461i)10-s + (0.250 + 2.02i)11-s + (4.01 − 2.20i)12-s + (0.118 − 4.84i)13-s + (0.190 − 0.146i)14-s + (−0.783 − 0.643i)15-s + (0.285 + 3.98i)16-s + (−2.35 − 2.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.930 − 0.366i)2-s + (0.414 − 1.25i)3-s + (0.731 + 0.681i)4-s + (0.0713 − 0.184i)5-s + (−0.844 + 1.01i)6-s + (−0.0330 + 0.0550i)7-s + (−0.431 − 0.902i)8-s + (−0.598 − 0.443i)9-s + (−0.134 + 0.145i)10-s + (0.0754 + 0.611i)11-s + (1.15 − 0.635i)12-s + (0.0329 − 1.34i)13-s + (0.0509 − 0.0391i)14-s + (−0.202 − 0.166i)15-s + (0.0713 + 0.997i)16-s + (−0.571 − 0.696i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.250780 - 0.904209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.250780 - 0.904209i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 + 0.517i)T \) |
| good | 3 | \( 1 + (-0.717 + 2.17i)T + (-2.40 - 1.78i)T^{2} \) |
| 5 | \( 1 + (-0.159 + 0.413i)T + (-3.70 - 3.35i)T^{2} \) |
| 7 | \( 1 + (0.0873 - 0.145i)T + (-3.29 - 6.17i)T^{2} \) |
| 11 | \( 1 + (-0.250 - 2.02i)T + (-10.6 + 2.67i)T^{2} \) |
| 13 | \( 1 + (-0.118 + 4.84i)T + (-12.9 - 0.637i)T^{2} \) |
| 17 | \( 1 + (2.35 + 2.87i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (5.88 + 4.14i)T + (6.40 + 17.8i)T^{2} \) |
| 23 | \( 1 + (-3.79 + 8.02i)T + (-14.5 - 17.7i)T^{2} \) |
| 29 | \( 1 + (9.52 - 2.63i)T + (24.8 - 14.9i)T^{2} \) |
| 31 | \( 1 + (-6.48 + 1.29i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-1.65 - 7.35i)T + (-33.4 + 15.8i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.280i)T + (4.01 - 40.8i)T^{2} \) |
| 43 | \( 1 + (-0.816 + 1.62i)T + (-25.6 - 34.5i)T^{2} \) |
| 47 | \( 1 + (7.92 + 4.23i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (10.6 + 2.95i)T + (45.4 + 27.2i)T^{2} \) |
| 59 | \( 1 + (-8.97 + 0.220i)T + (58.9 - 2.89i)T^{2} \) |
| 61 | \( 1 + (3.45 + 2.97i)T + (8.95 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.93 + 0.142i)T + (66.2 - 9.83i)T^{2} \) |
| 71 | \( 1 + (-0.874 + 5.89i)T + (-67.9 - 20.6i)T^{2} \) |
| 73 | \( 1 + (-4.92 - 8.21i)T + (-34.4 + 64.3i)T^{2} \) |
| 79 | \( 1 + (-8.52 - 2.58i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (1.18 - 5.27i)T + (-75.0 - 35.4i)T^{2} \) |
| 89 | \( 1 + (-4.56 - 9.65i)T + (-56.4 + 68.7i)T^{2} \) |
| 97 | \( 1 + (-8.38 - 5.60i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.58888890039363257070056224522, −9.475214669336593560030961632914, −8.594803259121249404408412402409, −7.995525211187999992549446198848, −6.98527896271057737710355025499, −6.51481227450104608669304035931, −4.77965195692964365254955053747, −2.96792929623718256739709808727, −2.13923170733216417657292209624, −0.70598545291622532799284384039,
1.93759016132791529421249489470, 3.50622584108399388651916862484, 4.53299237546491571233225854875, 5.88618173064066657686584756928, 6.72295601543647077350432203314, 7.948942159727869479760010936922, 8.894637862464339139930410645156, 9.328604455883849368114553452364, 10.22795572647306144841677971653, 10.94855885996800378147311098457
