L(s) = 1 | + 2-s + (−1.03 + 1.38i)3-s + 4-s + (−1.98 − 1.02i)5-s + (−1.03 + 1.38i)6-s + 8-s + (−0.844 − 2.87i)9-s + (−1.98 − 1.02i)10-s − 1.27i·11-s + (−1.03 + 1.38i)12-s + 1.32·13-s + (3.48 − 1.69i)15-s + 16-s + 5.76i·17-s + (−0.844 − 2.87i)18-s + 2.44i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.599 + 0.800i)3-s + 0.5·4-s + (−0.888 − 0.457i)5-s + (−0.423 + 0.566i)6-s + 0.353·8-s + (−0.281 − 0.959i)9-s + (−0.628 − 0.323i)10-s − 0.385i·11-s + (−0.299 + 0.400i)12-s + 0.366·13-s + (0.899 − 0.437i)15-s + 0.250·16-s + 1.39i·17-s + (−0.199 − 0.678i)18-s + 0.560i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.368946837\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.368946837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.03 - 1.38i)T \) |
| 5 | \( 1 + (1.98 + 1.02i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.27iT - 11T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 - 5.76iT - 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 - 0.680T + 23T^{2} \) |
| 29 | \( 1 - 6.12iT - 29T^{2} \) |
| 31 | \( 1 - 6.94iT - 31T^{2} \) |
| 37 | \( 1 - 4.67iT - 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 - 1.87iT - 43T^{2} \) |
| 47 | \( 1 + 7.97iT - 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 + 4.22iT - 61T^{2} \) |
| 67 | \( 1 - 12.9iT - 67T^{2} \) |
| 71 | \( 1 - 5.09iT - 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 3.33iT - 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 2.90T + 97T^{2} \) |
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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
−10.03696056563615146450156086609, −8.666475738710192321288766771347, −8.422285175884010546370565952882, −7.09476087310519722948575299888, −6.30583437309184647603644387446, −5.38780975330212828188656312124, −4.75257652731594510340431498243, −3.73480337139358534390931073732, −3.35934195464593466436559831765, −1.34905110037378811107947335786,
0.49352689951166706363851152888, 2.16237838010484063025270230784, 3.11697266181150207424076229426, 4.30941059559086671057519559984, 5.05083989621447803832711745120, 6.07347015100686764777009250251, 6.82594730018927938882650265948, 7.46293884914422233164083362861, 8.051760445401293811978551236282, 9.255601259623333437397155569481