L(s) = 1 | + (0.258 − 0.965i)2-s + 3-s + (0.965 − 0.258i)5-s + (0.258 − 0.965i)6-s + (−0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s − i·10-s + (−0.707 + 0.707i)11-s + (−0.499 + 0.866i)14-s + (0.965 − 0.258i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.707 − 0.707i)19-s + (−0.965 − 0.258i)21-s + (0.500 + 0.866i)22-s + (−0.866 + 0.5i)23-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + 3-s + (0.965 − 0.258i)5-s + (0.258 − 0.965i)6-s + (−0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s − i·10-s + (−0.707 + 0.707i)11-s + (−0.499 + 0.866i)14-s + (0.965 − 0.258i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.707 − 0.707i)19-s + (−0.965 − 0.258i)21-s + (0.500 + 0.866i)22-s + (−0.866 + 0.5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.820378829\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.820378829\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.965 + 0.258i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 3 | \( 1 - T + T^{2} \) |
| 5 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−9.803852479506050180499786943039, −9.381253296930247716360372169076, −8.247375236743790221635962853618, −7.44390109875117076567878055808, −6.47685197368399409438061111307, −5.47634359702900245078707687958, −4.21357518462696140688316014814, −3.28496397545506401662201132979, −2.54703483499142868448390636643, −1.71547530436726376240172844413,
2.17650547640900464162521045918, 2.79940455946519419911530112165, 3.99199745682297909546108246992, 5.64902910516702099971836409533, 5.80430558361857796543991863387, 6.68841626444651487081236432640, 7.76893229219265224895973141473, 8.259427692925799133327639696003, 9.234931971276750937654900041581, 9.971568626814928412866224014966