L(s) = 1 | + (0.222 − 0.385i)2-s + (0.400 + 0.694i)4-s + (−0.5 − 0.866i)7-s + 0.801·8-s + (−0.5 − 0.866i)9-s + (0.900 − 1.56i)11-s − 0.445·14-s + (−0.222 + 0.385i)16-s − 0.445·18-s + (−0.400 − 0.694i)22-s + (−0.623 + 1.07i)23-s + 25-s + (0.400 − 0.694i)28-s + (0.222 − 0.385i)29-s + (0.5 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.222 − 0.385i)2-s + (0.400 + 0.694i)4-s + (−0.5 − 0.866i)7-s + 0.801·8-s + (−0.5 − 0.866i)9-s + (0.900 − 1.56i)11-s − 0.445·14-s + (−0.222 + 0.385i)16-s − 0.445·18-s + (−0.400 − 0.694i)22-s + (−0.623 + 1.07i)23-s + 25-s + (0.400 − 0.694i)28-s + (0.222 − 0.385i)29-s + (0.5 + 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.276615136\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276615136\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.960507305742466820472188774415, −8.940911194272316541499343285519, −8.336342966554610163845569997812, −7.28104054035804095777275513574, −6.54748514943149056428589600579, −5.82625003780486837431086719977, −4.28456390588928114035296435740, −3.49591029498450966710325083184, −3.00084110576151595342055943756, −1.18369570412579761504192909885,
1.82360317619439863923624021570, 2.62450773620482330812293097375, 4.26760307831401861117243192288, 5.09246166150546607251209583061, 5.87544200087532436915344533814, 6.74596833958292934145114791207, 7.31324349311347439811826273330, 8.512123381592811528035069774963, 9.250147818954339808087709946249, 10.14385895347597702847102862804