| L(s) = 1 | + (1.32 − 0.484i)2-s + (−0.256 + 0.0643i)3-s + (1.53 − 1.28i)4-s + (2.70 + 1.28i)5-s + (−0.310 + 0.209i)6-s + (0.0816 + 0.269i)7-s + (1.41 − 2.45i)8-s + (−2.58 + 1.38i)9-s + (4.21 + 0.390i)10-s + (−3.27 + 0.486i)11-s + (−0.310 + 0.429i)12-s + (0.185 − 0.517i)13-s + (0.238 + 0.318i)14-s + (−0.778 − 0.154i)15-s + (0.688 − 3.94i)16-s + (2.69 − 0.536i)17-s + ⋯ |
| L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.148 + 0.0371i)3-s + (0.765 − 0.643i)4-s + (1.21 + 0.572i)5-s + (−0.126 + 0.0857i)6-s + (0.0308 + 0.101i)7-s + (0.499 − 0.866i)8-s + (−0.861 + 0.460i)9-s + (1.33 + 0.123i)10-s + (−0.988 + 0.146i)11-s + (−0.0896 + 0.123i)12-s + (0.0513 − 0.143i)13-s + (0.0638 + 0.0850i)14-s + (−0.200 − 0.0399i)15-s + (0.172 − 0.985i)16-s + (0.654 − 0.130i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.19992 - 0.356826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.19992 - 0.356826i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 + 0.484i)T \) |
| good | 3 | \( 1 + (0.256 - 0.0643i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-2.70 - 1.28i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (-0.0816 - 0.269i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (3.27 - 0.486i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-0.185 + 0.517i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-2.69 + 0.536i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (2.75 + 2.49i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.52 + 0.150i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (0.870 + 1.17i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (6.89 + 2.85i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.295 - 6.02i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (4.03 - 4.91i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (1.64 - 6.58i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (2.74 + 1.83i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (4.00 - 5.40i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-2.64 - 7.40i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (1.33 - 2.22i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-7.38 - 4.42i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-6.65 + 12.4i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-3.77 + 12.4i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-1.45 - 2.17i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.239 + 4.87i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-18.7 - 1.84i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (3.53 - 8.54i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.98941257274449046503744007486, −10.94063472502900115386602540825, −10.41599153958290978046976297309, −9.433987706456330279565500781459, −7.87887421810472970899028750031, −6.56391234251081891745156052259, −5.68676312673218929107753033254, −4.94793709392269542953453334672, −3.06864951258779858251556026468, −2.16060556477653181431590723873,
2.10661315348401729176504428231, 3.56719197214625165219578561811, 5.29956587918049620839471121118, 5.61184353130966993785691649998, 6.76804741210315679033609688537, 8.098069551843709777866942944864, 9.059941227045848879399612132286, 10.33973908544930886309375163867, 11.23327109967176438562170530215, 12.50283703808973406984194644347
