| L(s) = 1 | + i·5-s + 2.76i·7-s − 2.16i·11-s + (−0.167 − 3.60i)13-s − 5.03i·19-s + 4.93·23-s − 25-s + 4.43·29-s + 3.37i·31-s − 2.76·35-s − 3.97i·37-s − 1.66i·41-s + 9.80·43-s − 11.6i·47-s − 0.665·49-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 1.04i·7-s − 0.653i·11-s + (−0.0463 − 0.998i)13-s − 1.15i·19-s + 1.02·23-s − 0.200·25-s + 0.823·29-s + 0.605i·31-s − 0.468·35-s − 0.653i·37-s − 0.260i·41-s + 1.49·43-s − 1.69i·47-s − 0.0951·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.783866162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783866162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (0.167 + 3.60i)T \) |
| good | 7 | \( 1 - 2.76iT - 7T^{2} \) |
| 11 | \( 1 + 2.16iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.03iT - 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 3.37iT - 31T^{2} \) |
| 37 | \( 1 + 3.97iT - 37T^{2} \) |
| 41 | \( 1 + 1.66iT - 41T^{2} \) |
| 43 | \( 1 - 9.80T + 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 8.16iT - 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 7.10iT - 67T^{2} \) |
| 71 | \( 1 - 16.1iT - 71T^{2} \) |
| 73 | \( 1 - 15.9iT - 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 3.97iT - 83T^{2} \) |
| 89 | \( 1 - 7.94iT - 89T^{2} \) |
| 97 | \( 1 + 0.462iT - 97T^{2} \) |
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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
−8.751800327361507420080530800000, −8.529364664912121467049746495069, −7.33937414515041251213328730309, −6.75745138819917264158408087677, −5.63108910034262353480548789549, −5.36759018431193465145157145202, −4.09743838947792036830666820661, −2.92222282986635740207571752738, −2.52274854458130399872027337840, −0.813781680327905000814484323365,
0.952613530724991696520135973400, 1.98572615896784423909858631651, 3.33049680801616701805898349897, 4.34210460240264325268139481714, 4.71414261574394912987728733569, 5.94807882913201412714265774558, 6.71802740336617988140173459727, 7.50752275492634223665112221328, 8.068027724007890703152461139574, 9.176729463716614888138972594897
