L(s) = 1 | + (−0.960 − 0.278i)2-s + (0.845 + 0.534i)4-s + (−0.200 + 0.979i)5-s + (−0.663 − 0.748i)8-s + (0.721 − 0.692i)9-s + (0.464 − 0.885i)10-s + (−0.948 + 0.316i)13-s + (0.428 + 0.903i)16-s + (1.74 − 0.871i)17-s + (−0.885 + 0.464i)18-s + (−0.692 + 0.721i)20-s + (−0.919 − 0.391i)25-s + (0.999 − 0.0402i)26-s + (1.53 + 0.444i)29-s + (−0.160 − 0.987i)32-s + ⋯ |
L(s) = 1 | + (−0.960 − 0.278i)2-s + (0.845 + 0.534i)4-s + (−0.200 + 0.979i)5-s + (−0.663 − 0.748i)8-s + (0.721 − 0.692i)9-s + (0.464 − 0.885i)10-s + (−0.948 + 0.316i)13-s + (0.428 + 0.903i)16-s + (1.74 − 0.871i)17-s + (−0.885 + 0.464i)18-s + (−0.692 + 0.721i)20-s + (−0.919 − 0.391i)25-s + (0.999 − 0.0402i)26-s + (1.53 + 0.444i)29-s + (−0.160 − 0.987i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8499977481\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8499977481\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.960 + 0.278i)T \) |
| 5 | \( 1 + (0.200 - 0.979i)T \) |
| 13 | \( 1 + (0.948 - 0.316i)T \) |
good | 3 | \( 1 + (-0.721 + 0.692i)T^{2} \) |
| 7 | \( 1 + (-0.799 - 0.600i)T^{2} \) |
| 11 | \( 1 + (0.999 - 0.0402i)T^{2} \) |
| 17 | \( 1 + (-1.74 + 0.871i)T + (0.600 - 0.799i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-1.53 - 0.444i)T + (0.845 + 0.534i)T^{2} \) |
| 31 | \( 1 + (0.663 + 0.748i)T^{2} \) |
| 37 | \( 1 + (0.158 + 0.0258i)T + (0.948 + 0.316i)T^{2} \) |
| 41 | \( 1 + (0.360 + 0.895i)T + (-0.721 + 0.692i)T^{2} \) |
| 43 | \( 1 + (-0.316 - 0.948i)T^{2} \) |
| 47 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 53 | \( 1 + (-0.0539 + 0.891i)T + (-0.992 - 0.120i)T^{2} \) |
| 59 | \( 1 + (-0.774 + 0.632i)T^{2} \) |
| 61 | \( 1 + (0.156 - 0.768i)T + (-0.919 - 0.391i)T^{2} \) |
| 67 | \( 1 + (0.428 - 0.903i)T^{2} \) |
| 71 | \( 1 + (-0.960 - 0.278i)T^{2} \) |
| 73 | \( 1 + (-0.239 + 0.970i)T + (-0.885 - 0.464i)T^{2} \) |
| 79 | \( 1 + (0.568 + 0.822i)T^{2} \) |
| 83 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 89 | \( 1 + (-1.90 + 0.509i)T + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (0.150 - 1.86i)T + (-0.987 - 0.160i)T^{2} \) |
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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
−8.976680929436227892899020348038, −7.891134494305481838090838128163, −7.38713928413915608234694127972, −6.85614996968876377767259048219, −6.14531237770331492922373484565, −4.97949565843898104118644413561, −3.77317526447900208734216252737, −3.11853832622040406407414336322, −2.26738602351626408833097834265, −0.960280891666653148879634361298,
0.980035724361933404560283008465, 1.87984618026144862247736540774, 3.04617233456552128591703474975, 4.32110099720510172767588620456, 5.14309491759355151156542521728, 5.74564926591146737027688389589, 6.75725542450494235986884790857, 7.65937547472248972866631259656, 7.987041170635673276730382597394, 8.618650571208953761603907470597