| L(s) = 1 | − i·2-s − 4-s − 3.60i·5-s + 1.10i·7-s + i·8-s − 3.60·10-s + 2.35i·11-s + 1.10·14-s + 16-s + 5.96·17-s − 0.911i·19-s + 3.60i·20-s + 2.35·22-s + 3.38·23-s − 7.98·25-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.61i·5-s + 0.419i·7-s + 0.353i·8-s − 1.13·10-s + 0.710i·11-s + 0.296·14-s + 0.250·16-s + 1.44·17-s − 0.209i·19-s + 0.805i·20-s + 0.502·22-s + 0.705·23-s − 1.59·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.685254825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.685254825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.60iT - 5T^{2} \) |
| 7 | \( 1 - 1.10iT - 7T^{2} \) |
| 11 | \( 1 - 2.35iT - 11T^{2} \) |
| 17 | \( 1 - 5.96T + 17T^{2} \) |
| 19 | \( 1 + 0.911iT - 19T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 + 8.49iT - 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 7.18iT - 41T^{2} \) |
| 43 | \( 1 - 0.515T + 43T^{2} \) |
| 47 | \( 1 + 6.98iT - 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 + 10.1iT - 59T^{2} \) |
| 61 | \( 1 + 0.439T + 61T^{2} \) |
| 67 | \( 1 + 2.14iT - 67T^{2} \) |
| 71 | \( 1 - 0.615iT - 71T^{2} \) |
| 73 | \( 1 - 6.32iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 0.911iT - 83T^{2} \) |
| 89 | \( 1 + 3.75iT - 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−8.526787140012080959869427783857, −8.004298292683919673737549713379, −7.06121571431104173999428407201, −5.76098085675854697040575154647, −5.24176371627898152662974912409, −4.53716738054647788713314899516, −3.74018401203051118316850230193, −2.56514887054693975881165220339, −1.54821710628673489351598543893, −0.62990410583433074933006344720,
1.15070178773017464160242653461, 2.86602282529951307294569035982, 3.30577722963224612401676361697, 4.29389173115548140476434215572, 5.45788621746613268574449227377, 6.06680641287520256933375771147, 6.86315943536191662314844371087, 7.34338486509996553470676692471, 8.029267647504042118271898359810, 8.861020552517146852524143233944
