L(s) = 1 | − i·3-s + (1 − 2i)5-s − i·7-s − 9-s − 6·11-s − 6i·13-s + (−2 − i)15-s + 2·19-s − 21-s − 8i·23-s + (−3 − 4i)25-s + i·27-s + 10·29-s − 2·31-s + 6i·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.447 − 0.894i)5-s − 0.377i·7-s − 0.333·9-s − 1.80·11-s − 1.66i·13-s + (−0.516 − 0.258i)15-s + 0.458·19-s − 0.218·21-s − 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.192i·27-s + 1.85·29-s − 0.359·31-s + 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9843913871\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9843913871\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.100198088762637971247728741586, −7.78013129243434975082598804130, −6.61330454811213264582109507050, −5.92179092235838098545030041937, −5.01251073711345163935679115923, −4.73215144914997529827776690150, −3.07507896326722737879049119379, −2.58150475567522762056755070276, −1.20126884126363522542248350808, −0.29949943485577545580429800866,
1.86726545830743380500800015427, 2.64803013172638326691747812633, 3.45400992377609536384687406305, 4.45376067954748012345844059185, 5.44236370150368742515792824190, 5.77498393794199755421930674200, 7.00786101523006531554763313414, 7.32889798147739994935466139538, 8.453199198244193886678299260433, 9.068146146729537114272930467029