| L(s) = 1 | + (0.510 − 1.31i)2-s + (−0.891 + 0.453i)3-s + (−1.47 − 1.34i)4-s + (−1.27 − 1.83i)5-s + (0.143 + 1.40i)6-s + (−3.58 + 3.58i)7-s + (−2.53 + 1.26i)8-s + (0.587 − 0.809i)9-s + (−3.07 + 0.747i)10-s + (2.14 + 2.95i)11-s + (1.92 + 0.528i)12-s + (−2.29 + 0.363i)13-s + (2.89 + 6.55i)14-s + (1.97 + 1.05i)15-s + (0.371 + 3.98i)16-s + (0.0789 − 0.154i)17-s + ⋯ |
| L(s) = 1 | + (0.361 − 0.932i)2-s + (−0.514 + 0.262i)3-s + (−0.739 − 0.673i)4-s + (−0.571 − 0.820i)5-s + (0.0586 + 0.574i)6-s + (−1.35 + 1.35i)7-s + (−0.894 + 0.446i)8-s + (0.195 − 0.269i)9-s + (−0.971 + 0.236i)10-s + (0.647 + 0.891i)11-s + (0.556 + 0.152i)12-s + (−0.636 + 0.100i)13-s + (0.774 + 1.75i)14-s + (0.508 + 0.272i)15-s + (0.0929 + 0.995i)16-s + (0.0191 − 0.0375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0131806 + 0.0209050i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0131806 + 0.0209050i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.510 + 1.31i)T \) |
| 3 | \( 1 + (0.891 - 0.453i)T \) |
| 5 | \( 1 + (1.27 + 1.83i)T \) |
| good | 7 | \( 1 + (3.58 - 3.58i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.14 - 2.95i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.29 - 0.363i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.0789 + 0.154i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (2.48 + 7.63i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.36 + 0.532i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (3.85 + 1.25i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.48 - 2.10i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.376 + 2.37i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (1.04 + 0.759i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (1.11 + 1.11i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.44 + 4.79i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-5.47 - 10.7i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-8.45 - 6.14i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.26 + 2.37i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (6.44 + 3.28i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (2.21 + 0.718i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.489 - 3.09i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (1.57 - 4.83i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.71 + 9.25i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (0.271 + 0.373i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (7.98 - 4.06i)T + (57.0 - 78.4i)T^{2} \) |
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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
−12.08812876453072323473773805973, −11.46966761990801667584189386106, −10.14539448914964844088889673412, −9.200950701138527121347836704487, −8.948098525787059580286581766117, −6.96374130158627541188866644292, −5.75867381577272839893907846171, −4.84114346069107172976856835529, −3.77920791806510382354198778413, −2.30393479936043479970122548508,
0.01645927045901646354289387491, 3.48054756067442578546943737375, 4.00354579251173353615379543282, 5.83726876206451959686484656446, 6.56744618097485389959374851873, 7.27993865955397620268376125907, 8.128903475959219952450313499134, 9.679591490177586831743563413670, 10.43113429132068292615571704982, 11.58218927346900077100247930283
