| L(s) = 1 | + (1.22 − 0.707i)2-s + (−2.44 − 1.73i)3-s + (0.999 − 1.73i)4-s + (−4.22 − 0.389i)6-s + (3.17 + 5.49i)7-s − 2.82i·8-s + (2.99 + 8.48i)9-s + (8.17 − 4.71i)11-s + (−5.44 + 2.51i)12-s + (9.84 − 17.0i)13-s + (7.77 + 4.48i)14-s + (−2.00 − 3.46i)16-s − 1.90i·17-s + (9.67 + 8.27i)18-s + 4.69·19-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.816 − 0.577i)3-s + (0.249 − 0.433i)4-s + (−0.704 − 0.0648i)6-s + (0.453 + 0.785i)7-s − 0.353i·8-s + (0.333 + 0.942i)9-s + (0.743 − 0.429i)11-s + (−0.454 + 0.209i)12-s + (0.757 − 1.31i)13-s + (0.555 + 0.320i)14-s + (−0.125 − 0.216i)16-s − 0.112i·17-s + (0.537 + 0.459i)18-s + 0.247·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00922 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00922 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.027542705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.027542705\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (2.44 + 1.73i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-3.17 - 5.49i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.17 + 4.71i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-9.84 + 17.0i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 1.90iT - 289T^{2} \) |
| 19 | \( 1 - 4.69T + 361T^{2} \) |
| 23 | \( 1 + (8.17 + 4.71i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (2.84 - 1.64i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-20.5 + 35.5i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 17.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (53.5 + 30.9i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-0.477 - 0.826i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-12.2 + 7.05i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 9.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-79.2 - 45.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-37.5 - 65.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-15.4 + 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 96.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (14.8 + 25.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-76.1 + 43.9i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-47.9 - 83.0i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.91441083960441478339431832145, −10.12361231443348872382669704366, −8.743280569702237127370339656592, −7.85163691183019772464600475369, −6.59194322108848878462994657230, −5.77320652187322000761771959248, −5.12507115019193949456204310419, −3.70949498728045957766961427209, −2.25539259001146811665810036940, −0.876263518207804846469554753892,
1.44012242179938115738635517860, 3.67671837555965902666453376239, 4.33351358563758517051952525070, 5.25191913101009285973581735384, 6.52888991105375413515133053875, 6.93428324872963338170673153307, 8.341396549002316989717883630724, 9.416332360491096018788192194961, 10.35759099570765820841022328276, 11.34399013348986047481904036817
