L(s) = 1 | − 1.73i·3-s + 5-s + 2.44·7-s − 5.65i·13-s − 1.73i·15-s + 4.24i·17-s − 4.24i·21-s − 8.66i·23-s − 4·25-s − 5.19i·27-s + 5.65i·29-s + 1.73i·31-s + 2.44·35-s + 5·37-s − 9.79·39-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + 0.447·5-s + 0.925·7-s − 1.56i·13-s − 0.447i·15-s + 1.02i·17-s − 0.925i·21-s − 1.80i·23-s − 0.800·25-s − 1.00i·27-s + 1.05i·29-s + 0.311i·31-s + 0.414·35-s + 0.821·37-s − 1.56·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0829 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0829 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.153214414\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.153214414\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 1.73iT - 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8.66iT - 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 12.1iT - 59T^{2} \) |
| 61 | \( 1 - 5.65iT - 61T^{2} \) |
| 67 | \( 1 + 5.19iT - 67T^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 - 7.34T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 + 17T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−8.672901557243941736704659438611, −8.098470001670413137778725504791, −7.57907435910970493063774121847, −6.58346833560840022942902010968, −5.92129999962561155097613226907, −5.06744832617590582560101058765, −4.07071741787120852921910169558, −2.72851533561087580445039356598, −1.83191393608043881656109789124, −0.840724524961024415341250550791,
1.46076762786418715106388083802, 2.47551004291015946916148887625, 3.93173902905029603075225992029, 4.38179055370844287014344061657, 5.27092012538120295092150243821, 6.01127146126206876283211027968, 7.21311193701810972212431640567, 7.74311835399642303499359557922, 8.975333649224235834957446558097, 9.466406420687729479127564874605