| L(s) = 1 | + 3i·3-s + (−13.9 − 13.9i)5-s + (−3.70 − 3.70i)7-s − 9·9-s + (−47.9 − 47.9i)11-s + (−44.1 − 15.7i)13-s + (41.7 − 41.7i)15-s + 16.9i·17-s + (−4.73 + 4.73i)19-s + (11.1 − 11.1i)21-s + 185.·23-s + 262. i·25-s − 27i·27-s − 63.8·29-s + (−40.5 + 40.5i)31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−1.24 − 1.24i)5-s + (−0.199 − 0.199i)7-s − 0.333·9-s + (−1.31 − 1.31i)11-s + (−0.941 − 0.337i)13-s + (0.718 − 0.718i)15-s + 0.241i·17-s + (−0.0571 + 0.0571i)19-s + (0.115 − 0.115i)21-s + 1.68·23-s + 2.09i·25-s − 0.192i·27-s − 0.408·29-s + (−0.234 + 0.234i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3307810668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3307810668\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 13 | \( 1 + (44.1 + 15.7i)T \) |
| good | 5 | \( 1 + (13.9 + 13.9i)T + 125iT^{2} \) |
| 7 | \( 1 + (3.70 + 3.70i)T + 343iT^{2} \) |
| 11 | \( 1 + (47.9 + 47.9i)T + 1.33e3iT^{2} \) |
| 17 | \( 1 - 16.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (4.73 - 4.73i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 63.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + (40.5 - 40.5i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (153. - 153. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (-244. - 244. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + 204.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (163. + 163. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 - 745.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (356. + 356. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 - 633.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (254. - 254. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + (-638. + 638. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (239. - 239. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 638. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (718. - 718. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + (-34.5 + 34.5i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 + (346. + 346. i)T + 9.12e5iT^{2} \) |
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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
−10.50725966631045500068344850308, −9.451539836163258373860906969892, −8.480765148735276042535731333833, −8.097483765193630551363093771943, −7.04520526389190650890789091548, −5.36128012318726401206655712388, −5.01696668125361432446898972478, −3.82596777408987525572708149075, −2.93045639245907352224448789978, −0.76603826029790341702764620848,
0.13846497966558078461624681199, 2.29784367113032286279586174709, 3.00067990742151626101125611719, 4.33309276295377458738503233075, 5.40444966123950178795920752823, 6.95705693837906398530173065136, 7.20814593534771944902658298136, 7.86093280579458515320690953609, 9.107808214469886560916678329911, 10.21484050972902400651156485064
