L(s) = 1 | + (0.909 − 0.415i)2-s + (−0.755 − 0.654i)3-s + (0.654 − 0.755i)4-s + (0.989 + 0.142i)5-s + (−0.959 − 0.281i)6-s + (0.281 − 0.959i)8-s + (0.142 + 0.989i)9-s + (0.959 − 0.281i)10-s + (−0.989 + 0.142i)12-s + (−0.654 − 0.755i)15-s + (−0.142 − 0.989i)16-s + (−0.909 − 1.41i)17-s + (0.540 + 0.841i)18-s + (0.698 + 0.449i)19-s + (0.755 − 0.654i)20-s + ⋯ |
L(s) = 1 | + (0.909 − 0.415i)2-s + (−0.755 − 0.654i)3-s + (0.654 − 0.755i)4-s + (0.989 + 0.142i)5-s + (−0.959 − 0.281i)6-s + (0.281 − 0.959i)8-s + (0.142 + 0.989i)9-s + (0.959 − 0.281i)10-s + (−0.989 + 0.142i)12-s + (−0.654 − 0.755i)15-s + (−0.142 − 0.989i)16-s + (−0.909 − 1.41i)17-s + (0.540 + 0.841i)18-s + (0.698 + 0.449i)19-s + (0.755 − 0.654i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.702111750\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.702111750\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.909 + 0.415i)T \) |
| 3 | \( 1 + (0.755 + 0.654i)T \) |
| 5 | \( 1 + (-0.989 - 0.142i)T \) |
| 23 | \( 1 + (-0.281 - 0.959i)T \) |
good | 7 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 17 | \( 1 + (0.909 + 1.41i)T + (-0.415 + 0.909i)T^{2} \) |
| 19 | \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 31 | \( 1 + (1.37 - 1.19i)T + (0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 41 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + 1.91iT - T^{2} \) |
| 53 | \( 1 + (1.74 - 0.797i)T + (0.654 - 0.755i)T^{2} \) |
| 59 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (-0.425 + 0.368i)T + (0.142 - 0.989i)T^{2} \) |
| 67 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 71 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 79 | \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (-0.215 - 1.49i)T + (-0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 97 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
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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
−9.787090766530453380297860695431, −9.074689183829569506791279269075, −7.51777935441074721809665694092, −6.90987354410814484668949855303, −6.18588430074037569554844520408, −5.26872052822808582137330530897, −4.94013167194391144526728518004, −3.38279868236033452294047202971, −2.30189539363183973687411219214, −1.36635764966846977664107319026,
1.89033792249303889225798080602, 3.18837304230988450899383410123, 4.29678381926428295317233497240, 4.93507184461083917332952191345, 5.88553099208171364699703323145, 6.27942341867022313063179919794, 7.15282933722899709790493402674, 8.387006220362710576812231513238, 9.206860264918749177654162610385, 10.03350918506062950373787078471