L(s) = 1 | + (0.900 − 0.433i)2-s + (0.433 − 1.90i)3-s + (0.623 − 0.781i)4-s + (−0.433 − 1.90i)6-s + (0.974 + 0.222i)7-s + (0.222 − 0.974i)8-s + (−2.52 − 1.21i)9-s + (0.781 − 0.376i)11-s + (−1.21 − 1.52i)12-s + (−1.62 + 0.781i)13-s + (0.974 − 0.222i)14-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 2.80·18-s + (0.846 − 1.75i)21-s + (0.541 − 0.678i)22-s + ⋯ |
L(s) = 1 | + (0.900 − 0.433i)2-s + (0.433 − 1.90i)3-s + (0.623 − 0.781i)4-s + (−0.433 − 1.90i)6-s + (0.974 + 0.222i)7-s + (0.222 − 0.974i)8-s + (−2.52 − 1.21i)9-s + (0.781 − 0.376i)11-s + (−1.21 − 1.52i)12-s + (−1.62 + 0.781i)13-s + (0.974 − 0.222i)14-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 2.80·18-s + (0.846 − 1.75i)21-s + (0.541 − 0.678i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.588429190\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.588429190\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 7 | \( 1 + (-0.974 - 0.222i)T \) |
| 17 | \( 1 + (-0.623 - 0.781i)T \) |
good | 3 | \( 1 + (-0.433 + 1.90i)T + (-0.900 - 0.433i)T^{2} \) |
| 5 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.781 + 0.376i)T + (0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.541 - 0.678i)T + (-0.222 - 0.974i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 - 1.94T + T^{2} \) |
| 37 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 53 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 61 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1.21 + 1.52i)T + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 79 | \( 1 - 0.867T + T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.081504984212278044529573734169, −7.75324839177522967479045258714, −6.86660251904756859975224910785, −6.27991976741267735100037101896, −5.61122042115418984394762448297, −4.62648763225245566843811929897, −3.59506990121780447685758916552, −2.51101119212846296005406007100, −1.95040540858018553602810501150, −1.15215486302230537003789172583,
2.31574799004138625653225051349, 2.99783523918738259681014161286, 4.00354548623843837463063398171, 4.52041493401109337489911837641, 5.10883514961152540407019229825, 5.64132575359613991663494613238, 6.86196347446419068078581255035, 7.956049579301469208973203934570, 8.105874494272073648341098889915, 9.216942959544971842643543627453