| L(s) = 1 | + (−0.190 − 0.587i)2-s + (0.669 + 0.743i)3-s + (1.30 − 0.951i)4-s + (1.30 + 2.26i)5-s + (0.309 − 0.535i)6-s + (−0.313 − 2.98i)7-s + (−1.80 − 1.31i)8-s + (0.209 − 1.98i)9-s + (1.08 − 1.20i)10-s + (0.697 + 0.310i)11-s + (1.58 + 0.336i)12-s + (4.74 − 1.00i)13-s + (−1.69 + 0.754i)14-s + (−0.809 + 2.48i)15-s + (0.572 − 1.76i)16-s + (−0.215 + 0.0960i)17-s + ⋯ |
| L(s) = 1 | + (−0.135 − 0.415i)2-s + (0.386 + 0.429i)3-s + (0.654 − 0.475i)4-s + (0.585 + 1.01i)5-s + (0.126 − 0.218i)6-s + (−0.118 − 1.12i)7-s + (−0.639 − 0.464i)8-s + (0.0696 − 0.663i)9-s + (0.342 − 0.380i)10-s + (0.210 + 0.0936i)11-s + (0.456 + 0.0971i)12-s + (1.31 − 0.279i)13-s + (−0.452 + 0.201i)14-s + (−0.208 + 0.642i)15-s + (0.143 − 0.440i)16-s + (−0.0523 + 0.0232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94076 - 1.01341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94076 - 1.01341i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (0.190 + 0.587i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.669 - 0.743i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (-1.30 - 2.26i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.313 + 2.98i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-0.697 - 0.310i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-4.74 + 1.00i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (0.215 - 0.0960i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (4.89 + 1.03i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (4.42 + 3.21i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.66 - 8.19i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.118 + 0.204i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.33 + 4.80i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (4.51 + 0.960i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (1.04 - 3.21i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (1.32 - 12.6i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-6.33 - 7.03i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 + (-2.11 - 3.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.00942 - 0.0896i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-7.82 - 3.48i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-2.73 + 3.03i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (5.16 - 3.75i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.28 + 3.11i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.30531956394194561475469236719, −9.297673939970015028719641409053, −8.432003919121442887247283010626, −7.02539892635800815045452144146, −6.57427776926365566635208077385, −5.89070818541003599667393443612, −4.18216296447816092995189040098, −3.42586768077915414588102727396, −2.45464289867226235506060072542, −1.08008172402387320784562462080,
1.73153472587432438911458436904, 2.38639132884310982688608931021, 3.79902916918042833384525830623, 5.17928479402615558481475258286, 6.05219932854445865011461569447, 6.58119420710718705136716331098, 8.023965801541747252326410428483, 8.327449240360088242417507686243, 8.967564447535449671962930565656, 9.921101023880239904961005200618
