| L(s) = 1 | + (1.84 − 0.493i)2-s + (−0.312 + 0.180i)4-s + (1.82 + 4.65i)5-s + (−6.63 − 2.22i)7-s + (−5.88 + 5.88i)8-s + (5.65 + 7.68i)10-s + (5.24 + 9.07i)11-s + (1.40 − 1.40i)13-s + (−13.3 − 0.822i)14-s + (−7.21 + 12.4i)16-s + (−6.55 + 24.4i)17-s + (−9.07 − 5.23i)19-s + (−1.40 − 1.12i)20-s + (14.1 + 14.1i)22-s + (7.89 + 29.4i)23-s + ⋯ |
| L(s) = 1 | + (0.921 − 0.246i)2-s + (−0.0780 + 0.0450i)4-s + (0.364 + 0.931i)5-s + (−0.948 − 0.317i)7-s + (−0.735 + 0.735i)8-s + (0.565 + 0.768i)10-s + (0.476 + 0.825i)11-s + (0.108 − 0.108i)13-s + (−0.952 − 0.0587i)14-s + (−0.450 + 0.780i)16-s + (−0.385 + 1.43i)17-s + (−0.477 − 0.275i)19-s + (−0.0703 − 0.0562i)20-s + (0.642 + 0.642i)22-s + (0.343 + 1.28i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.16201 + 1.34083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16201 + 1.34083i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.82 - 4.65i)T \) |
| 7 | \( 1 + (6.63 + 2.22i)T \) |
| good | 2 | \( 1 + (-1.84 + 0.493i)T + (3.46 - 2i)T^{2} \) |
| 11 | \( 1 + (-5.24 - 9.07i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-1.40 + 1.40i)T - 169iT^{2} \) |
| 17 | \( 1 + (6.55 - 24.4i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (9.07 + 5.23i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-7.89 - 29.4i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 55.6iT - 841T^{2} \) |
| 31 | \( 1 + (-8.12 - 14.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-14.8 + 3.96i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 28.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-3.17 + 3.17i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-1.71 + 0.458i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-56.9 - 15.2i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (57.7 - 33.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-6.16 + 10.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-28.5 + 106. i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 29.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-115. - 30.8i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (99.9 + 57.6i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (24.2 - 24.2i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (18.3 + 10.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-56.7 - 56.7i)T + 9.40e3iT^{2} \) |
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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
−11.84978679410061760831819878764, −10.87347785958193294067541273929, −9.928248188913970118488115977046, −9.094617838001291457491570656056, −7.68976806954360476558291229263, −6.50772591308052587101025319300, −5.86305386903525266061451388811, −4.29309209090461576619024428283, −3.50913355911353291974015235197, −2.27645717708638156411568458392,
0.62416695351378160128144242038, 2.87814688290744655888034146826, 4.16174338167790064237017691371, 5.16565423003988285896122614639, 6.05940881605254373707715114734, 6.86378648830428611186818171292, 8.713127927123556471543356441066, 9.122591406575725844279612107175, 10.09314200930015257711504552446, 11.48960779555982280428611004162
