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
−11.72724895325712875770700513256, −11.36776459707274273109767195165, −10.47116103304358765361233937875, −9.143082186176398772756017266982, −8.336179446136585275679656523623, −7.15223383202169394450726861653, −6.62660717050320494007325853197, −4.55473460642466664166867915291, −3.26408742373867991986140773146, −1.42289267192764187594513476245,
0.880913168175172710937651558449, 3.53690639685039804350203296328, 5.04425499950965648330743211493, 6.03415392374771693883846725245, 7.29025896509972765583669066602, 8.323122789195239361730858682367, 8.950559582849331348957200332307, 10.19597815651876057565212401366, 11.15163122979972349089000587513, 11.62366999291647155783141343373