L(s) = 1 | + (−0.309 − 0.951i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (−0.809 + 0.587i)9-s + 0.999·12-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.809 − 0.587i)20-s − 2·23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (−1.61 + 1.17i)31-s + (−0.309 − 0.951i)36-s + 0.999·45-s + (0.618 + 1.90i)47-s + (−0.309 + 0.951i)48-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (−0.809 + 0.587i)9-s + 0.999·12-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.809 − 0.587i)20-s − 2·23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (−1.61 + 1.17i)31-s + (−0.309 − 0.951i)36-s + 0.999·45-s + (0.618 + 1.90i)47-s + (−0.309 + 0.951i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1538604558\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1538604558\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + 2T + T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−9.436996876924458651129433406559, −8.724250555028151935912127266186, −7.943795214674795353093899515017, −7.65049717121541997325555262822, −6.76223078182398118698394662105, −5.75905844211031755709008676512, −4.78428214810834825470227494186, −3.92738743065864183146225175975, −2.97717148550644085183360690348, −1.64957419471006557370417787151,
0.11657590058233451959682657159, 2.14085894493375642298972644180, 3.58479043149398759032717981350, 4.13870364400120784210035557835, 5.06834472441158219404922787744, 5.90071889215554155210256010090, 6.54470293394561449733858120162, 7.67745148583229730935224138478, 8.504560777955781884350327921667, 9.398807897831135551076206588091