| L(s) = 1 | + (−2.26 − 0.480i)2-s + (1.74 − 0.447i)3-s + (3.05 + 1.35i)4-s + (0.0484 − 2.31i)5-s + (−4.15 + 0.174i)6-s + (1.45 + 0.507i)7-s + (−2.50 − 1.82i)8-s + (0.210 − 0.115i)9-s + (−1.22 + 5.20i)10-s + (1.18 − 1.21i)11-s + (5.93 + 1.00i)12-s + (0.0277 + 0.187i)13-s + (−3.05 − 1.84i)14-s + (−0.951 − 4.05i)15-s + (0.324 + 0.360i)16-s + (−1.41 − 1.22i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 − 0.339i)2-s + (1.00 − 0.258i)3-s + (1.52 + 0.679i)4-s + (0.0216 − 1.03i)5-s + (−1.69 + 0.0711i)6-s + (0.551 + 0.191i)7-s + (−0.887 − 0.644i)8-s + (0.0700 − 0.0385i)9-s + (−0.386 + 1.64i)10-s + (0.358 − 0.365i)11-s + (1.71 + 0.289i)12-s + (0.00769 + 0.0521i)13-s + (−0.816 − 0.494i)14-s + (−0.245 − 1.04i)15-s + (0.0811 + 0.0901i)16-s + (−0.343 − 0.296i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.628662 - 0.429743i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.628662 - 0.429743i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-0.437 + 12.2i)T \) |
| good | 2 | \( 1 + (2.26 + 0.480i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-1.74 + 0.447i)T + (2.62 - 1.44i)T^{2} \) |
| 5 | \( 1 + (-0.0484 + 2.31i)T + (-4.99 - 0.209i)T^{2} \) |
| 7 | \( 1 + (-1.45 - 0.507i)T + (5.48 + 4.34i)T^{2} \) |
| 11 | \( 1 + (-1.18 + 1.21i)T + (-0.230 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0277 - 0.187i)T + (-12.4 + 3.75i)T^{2} \) |
| 17 | \( 1 + (1.41 + 1.22i)T + (2.48 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.34 + 3.15i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.826 + 7.85i)T + (-22.4 + 4.78i)T^{2} \) |
| 29 | \( 1 + (-0.318 - 5.06i)T + (-28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (4.23 - 10.0i)T + (-21.6 - 22.1i)T^{2} \) |
| 37 | \( 1 + (1.12 - 0.894i)T + (8.44 - 36.0i)T^{2} \) |
| 41 | \( 1 + (-4.55 - 7.18i)T + (-17.4 + 37.0i)T^{2} \) |
| 43 | \( 1 + (-2.10 + 0.733i)T + (33.6 - 26.7i)T^{2} \) |
| 47 | \( 1 + (0.254 - 0.487i)T + (-26.8 - 38.5i)T^{2} \) |
| 53 | \( 1 + (1.00 + 1.21i)T + (-9.93 + 52.0i)T^{2} \) |
| 59 | \( 1 + (2.75 - 8.48i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.27 - 8.14i)T + (-52.1 + 31.5i)T^{2} \) |
| 67 | \( 1 + (0.471 + 0.259i)T + (35.9 + 56.5i)T^{2} \) |
| 71 | \( 1 + (10.0 - 8.69i)T + (10.3 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.41 + 7.42i)T + (-67.8 + 26.8i)T^{2} \) |
| 79 | \( 1 + (-11.7 - 1.48i)T + (76.5 + 19.6i)T^{2} \) |
| 83 | \( 1 + (-6.89 - 14.6i)T + (-52.9 + 63.9i)T^{2} \) |
| 89 | \( 1 + (-2.98 + 4.29i)T + (-31.0 - 83.4i)T^{2} \) |
| 97 | \( 1 + (-15.1 + 9.19i)T + (44.9 - 85.9i)T^{2} \) |
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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
−12.63040823054880560415183683778, −11.58287642882563098485750933072, −10.60484351343529789820109662543, −9.170366587828080321320114720658, −8.830573013008132527577862194271, −8.132815734468032958350747012233, −7.01817744574331337425760134041, −4.95866106536187223069203251627, −2.80041991022947431467118710748, −1.31946390668548783306942479289,
2.05825880719091221163859089285, 3.69675252213719522631893613280, 6.09662617897614116910516514892, 7.50663308877841850741140490608, 7.87617886831534910207107573995, 9.226320930704446428692896657937, 9.730750000748456660290209165142, 10.84329285245619651423227574734, 11.65252171456131837045559329854, 13.58057305954589817664574149680
