L(s) = 1 | + (−1.20 + 0.742i)2-s + (−0.707 + 0.707i)3-s + (0.898 − 1.78i)4-s + (−2.06 − 0.860i)5-s + (0.326 − 1.37i)6-s + 0.707·7-s + (0.244 + 2.81i)8-s − 1.00i·9-s + (3.12 − 0.495i)10-s + (1.79 − 1.79i)11-s + (0.628 + 1.89i)12-s + (3.86 − 3.86i)13-s + (−0.851 + 0.524i)14-s + (2.06 − 0.850i)15-s + (−2.38 − 3.21i)16-s − 0.244i·17-s + ⋯ |
L(s) = 1 | + (−0.851 + 0.524i)2-s + (−0.408 + 0.408i)3-s + (0.449 − 0.893i)4-s + (−0.922 − 0.384i)5-s + (0.133 − 0.561i)6-s + 0.267·7-s + (0.0863 + 0.996i)8-s − 0.333i·9-s + (0.987 − 0.156i)10-s + (0.542 − 0.542i)11-s + (0.181 + 0.548i)12-s + (1.07 − 1.07i)13-s + (−0.227 + 0.140i)14-s + (0.533 − 0.219i)15-s + (−0.596 − 0.802i)16-s − 0.0593i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.660409 - 0.0492920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.660409 - 0.0492920i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.20 - 0.742i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.06 + 0.860i)T \) |
good | 7 | \( 1 - 0.707T + 7T^{2} \) |
| 11 | \( 1 + (-1.79 + 1.79i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3.86 + 3.86i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.244iT - 17T^{2} \) |
| 19 | \( 1 + (-1.53 - 1.53i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + (4.89 + 4.89i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.60T + 31T^{2} \) |
| 37 | \( 1 + (8.47 + 8.47i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.12iT - 41T^{2} \) |
| 43 | \( 1 + (0.684 + 0.684i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.47iT - 47T^{2} \) |
| 53 | \( 1 + (-1.47 - 1.47i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.86 + 5.86i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.0537 - 0.0537i)T + 61iT^{2} \) |
| 67 | \( 1 + (7.85 - 7.85i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.08iT - 71T^{2} \) |
| 73 | \( 1 - 9.69T + 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 + (6.80 - 6.80i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.07iT - 89T^{2} \) |
| 97 | \( 1 + 1.39iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.62366999291647155783141343373, −11.15163122979972349089000587513, −10.19597815651876057565212401366, −8.950559582849331348957200332307, −8.323122789195239361730858682367, −7.29025896509972765583669066602, −6.03415392374771693883846725245, −5.04425499950965648330743211493, −3.53690639685039804350203296328, −0.880913168175172710937651558449,
1.42289267192764187594513476245, 3.26408742373867991986140773146, 4.55473460642466664166867915291, 6.62660717050320494007325853197, 7.15223383202169394450726861653, 8.336179446136585275679656523623, 9.143082186176398772756017266982, 10.47116103304358765361233937875, 11.36776459707274273109767195165, 11.72724895325712875770700513256