| L(s) = 1 | + (−1.5 − 2.59i)3-s + (−8.93 + 15.4i)5-s + (−2.26 + 18.3i)7-s + (−4.5 + 7.79i)9-s + (−5.69 − 9.86i)11-s − 13.0·13-s + 53.6·15-s + (−26.6 − 46.1i)17-s + (−21.2 + 36.7i)19-s + (51.1 − 21.6i)21-s + (76.0 − 131. i)23-s + (−97.2 − 168. i)25-s + 27·27-s + 186.·29-s + (−78.9 − 136. i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.799 + 1.38i)5-s + (−0.122 + 0.992i)7-s + (−0.166 + 0.288i)9-s + (−0.156 − 0.270i)11-s − 0.279·13-s + 0.922·15-s + (−0.379 − 0.658i)17-s + (−0.256 + 0.443i)19-s + (0.531 − 0.225i)21-s + (0.689 − 1.19i)23-s + (−0.777 − 1.34i)25-s + 0.192·27-s + 1.19·29-s + (−0.457 − 0.792i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2199166633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2199166633\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 + (2.26 - 18.3i)T \) |
| good | 5 | \( 1 + (8.93 - 15.4i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (5.69 + 9.86i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + (26.6 + 46.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (21.2 - 36.7i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-76.0 + 131. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 186.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (78.9 + 136. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (1.87 - 3.24i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 39.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 429.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-10.5 + 18.3i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (182. + 316. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (113. + 196. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (325. - 564. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-72.7 - 125. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 368.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (304. + 527. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-455. + 788. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 327.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-18.8 + 32.5i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 722.T + 9.12e5T^{2} \) |
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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
−10.99187601186652913285458397823, −10.09964491151114349661632052846, −8.755698226222863132672470043107, −7.84786717715272018171392469431, −6.84310858060681519659519330283, −6.20138086017586087200132511517, −4.83189731738616839282600616649, −3.23400973665991076149526686226, −2.35665490359158363450495122272, −0.088990211620743331542927790705,
1.22055465967484617982988449749, 3.51234074273613870301861129901, 4.51270985189381862652062854615, 5.12912404910969385854759179472, 6.68998256313254856640718438857, 7.76713682967958325221761314958, 8.664060004432749174962540787530, 9.563430555789800826625569259014, 10.56592120082583912335950651916, 11.41624877989302077823001451779
