L(s) = 1 | + (−0.587 + 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 − 1.53i)4-s + (−0.363 − 1.11i)5-s + (0.587 + 1.80i)6-s + (2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + 2.23·10-s + (0.951 − 0.309i)11-s − 2.61·12-s + (−0.309 + 0.951i)13-s + (−0.951 − 0.690i)15-s + (1.00 + 3.07i)16-s + (1.53 + 1.11i)18-s + (−0.951 + 2.92i)20-s + ⋯ |
L(s) = 1 | + (−0.587 + 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 − 1.53i)4-s + (−0.363 − 1.11i)5-s + (0.587 + 1.80i)6-s + (2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + 2.23·10-s + (0.951 − 0.309i)11-s − 2.61·12-s + (−0.309 + 0.951i)13-s + (−0.951 − 0.690i)15-s + (1.00 + 3.07i)16-s + (1.53 + 1.11i)18-s + (−0.951 + 2.92i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7063709875\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7063709875\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (-0.951 + 0.309i)T \) |
| 13 | \( 1 + (0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−11.65903088490401107567097856052, −9.825425779206595117927571125462, −9.138508017627489683637531519705, −8.599094381886237332205799290960, −7.902473348955992452557548767872, −6.93826688803787273115641422483, −6.25174836855119128107168195073, −4.91491735992342149852661728556, −3.99226320678547777366318435248, −1.29084557054643480212245870162,
2.01248883791131144715996958368, 3.16965587422446130209045635720, 3.67596515694170709200913260421, 4.88767612510308161591205278996, 7.05943520761874582306518574253, 8.097177583582272718559967699311, 8.841662869976046946851248807864, 9.971184197995965109976995318224, 10.16699107324248710804034250246, 11.18616036958467706017399684130