| L(s) = 1 | − 2-s + 4-s − 1.15i·5-s + i·7-s − 8-s + 1.15i·10-s + (−1.65 + 2.87i)11-s − 0.143i·13-s − i·14-s + 16-s − 5.46·17-s − 3.61i·19-s − 1.15i·20-s + (1.65 − 2.87i)22-s − 9.38i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.514i·5-s + 0.377i·7-s − 0.353·8-s + 0.363i·10-s + (−0.500 + 0.865i)11-s − 0.0397i·13-s − 0.267i·14-s + 0.250·16-s − 1.32·17-s − 0.828i·19-s − 0.257i·20-s + (0.353 − 0.612i)22-s − 1.95i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0915 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0915 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8456267947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8456267947\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (1.65 - 2.87i)T \) |
| good | 5 | \( 1 + 1.15iT - 5T^{2} \) |
| 13 | \( 1 + 0.143iT - 13T^{2} \) |
| 17 | \( 1 + 5.46T + 17T^{2} \) |
| 19 | \( 1 + 3.61iT - 19T^{2} \) |
| 23 | \( 1 + 9.38iT - 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 - 5.05T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.569iT - 43T^{2} \) |
| 47 | \( 1 + 5.48iT - 47T^{2} \) |
| 53 | \( 1 + 13.0iT - 53T^{2} \) |
| 59 | \( 1 - 1.86iT - 59T^{2} \) |
| 61 | \( 1 - 2.44iT - 61T^{2} \) |
| 67 | \( 1 + 8.26T + 67T^{2} \) |
| 71 | \( 1 + 7.08iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 10.8iT - 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 - 17.0iT - 89T^{2} \) |
| 97 | \( 1 + 8.25T + 97T^{2} \) |
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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
−9.201364089651073802163525156122, −8.622894051261686881075072857949, −8.072811587274943170387770098610, −6.77260944002279341613316418663, −6.52034538562843433668659828412, −4.94632776983389572945102616994, −4.58800251082719975566710949815, −2.85940085086066039818041804400, −2.05971908740900258467492098664, −0.46908269531051970855496617664,
1.21700932096258726403975334251, 2.62427708949112133697426350380, 3.47545801673812543271281902613, 4.71373402870495323186708991087, 5.88058185078055300703584939765, 6.61465020465729566616965383520, 7.41106028123558231656926039771, 8.231003981431413045851870741218, 8.864378283206896160177667696947, 9.877679351194010397048371625743
