L(s) = 1 | − 2.34·3-s − 2.23i·5-s + 7.61i·7-s − 3.48·9-s + 12.3i·11-s − 13.0·13-s + 5.25i·15-s + 9.13i·17-s + 14.4i·19-s − 17.8i·21-s + (−22.5 − 4.51i)23-s − 5.00·25-s + 29.3·27-s − 21.2·29-s − 36.8·31-s + ⋯ |
L(s) = 1 | − 0.782·3-s − 0.447i·5-s + 1.08i·7-s − 0.387·9-s + 1.12i·11-s − 1.00·13-s + 0.350i·15-s + 0.537i·17-s + 0.760i·19-s − 0.851i·21-s + (−0.980 − 0.196i)23-s − 0.200·25-s + 1.08·27-s − 0.733·29-s − 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1589340312\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1589340312\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (22.5 + 4.51i)T \) |
good | 3 | \( 1 + 2.34T + 9T^{2} \) |
| 7 | \( 1 - 7.61iT - 49T^{2} \) |
| 11 | \( 1 - 12.3iT - 121T^{2} \) |
| 13 | \( 1 + 13.0T + 169T^{2} \) |
| 17 | \( 1 - 9.13iT - 289T^{2} \) |
| 19 | \( 1 - 14.4iT - 361T^{2} \) |
| 29 | \( 1 + 21.2T + 841T^{2} \) |
| 31 | \( 1 + 36.8T + 961T^{2} \) |
| 37 | \( 1 + 56.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 70.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 70.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 77.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 82.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 23.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 118. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 69.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 25.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 28.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 69.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 45.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 74.4iT - 9.40e3T^{2} \) |
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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
−9.157245752801088783595965476656, −7.920312264268806116356175902866, −7.39904205909868513432297156584, −6.02205418564666964177225860700, −5.80260924593989465641462548506, −4.87683539408424065269489482366, −4.07169893528543230475323336693, −2.59144809145806722591827418620, −1.77482099324541576795625107471, −0.06213670297023895847672683568,
0.77536027375349841428228559625, 2.43294267574070446948248919940, 3.43983153943960559490126991134, 4.38828199167207874126623035821, 5.40743496085203061482130835929, 5.99151216769592384647105729084, 7.02758579883136396270205245696, 7.43473775945244912099246211089, 8.482328889496263576275531963085, 9.376446027668317402364347872796