L(s) = 1 | + i·2-s + 3.12i·3-s − 4-s − 2.20i·5-s − 3.12·6-s − i·8-s − 6.77·9-s + 2.20·10-s + (−1.49 + 2.96i)11-s − 3.12i·12-s − 1.45·13-s + 6.88·15-s + 16-s − 7.60·17-s − 6.77i·18-s − 0.180·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.80i·3-s − 0.5·4-s − 0.985i·5-s − 1.27·6-s − 0.353i·8-s − 2.25·9-s + 0.696·10-s + (−0.449 + 0.893i)11-s − 0.902i·12-s − 0.404·13-s + 1.77·15-s + 0.250·16-s − 1.84·17-s − 1.59i·18-s − 0.0414·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09389201200\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09389201200\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (1.49 - 2.96i)T \) |
good | 3 | \( 1 - 3.12iT - 3T^{2} \) |
| 5 | \( 1 + 2.20iT - 5T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 + 0.180T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 + 4.45iT - 29T^{2} \) |
| 31 | \( 1 + 8.54iT - 31T^{2} \) |
| 37 | \( 1 - 1.50T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 1.58iT - 43T^{2} \) |
| 47 | \( 1 + 0.545iT - 47T^{2} \) |
| 53 | \( 1 - 4.83T + 53T^{2} \) |
| 59 | \( 1 - 6.18iT - 59T^{2} \) |
| 61 | \( 1 + 5.72T + 61T^{2} \) |
| 67 | \( 1 + 2.98T + 67T^{2} \) |
| 71 | \( 1 + 7.57T + 71T^{2} \) |
| 73 | \( 1 + 9.67T + 73T^{2} \) |
| 79 | \( 1 - 6.91iT - 79T^{2} \) |
| 83 | \( 1 + 5.84T + 83T^{2} \) |
| 89 | \( 1 - 16.1iT - 89T^{2} \) |
| 97 | \( 1 + 11.3iT - 97T^{2} \) |
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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
−10.23578067292465654793631101562, −9.600554577455351139883649847555, −8.955602404093265492117337413035, −8.416663994763515443821492731244, −7.28817528865348153289819739835, −6.06795709612365832913711192768, −5.11725283946306581158842757646, −4.57170771099171190058853440613, −4.06276665467721197613756287469, −2.50653422115915742205488504242,
0.03999687251798425386462523067, 1.56976949074715005063296826491, 2.62900331074328782307552181694, 3.19483576317159034464258206249, 4.91506777228892041352407360853, 6.07480603015160603525659126232, 6.82320221125774771537372902818, 7.34990602001853676278400265354, 8.487142622316923217392837826218, 8.895434505381250264075462926618