L(s) = 1 | + (−0.809 − 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 + 0.951i)9-s − 12-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)20-s + 2·23-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (0.618 + 1.90i)31-s + (0.809 + 0.587i)36-s − 45-s + (1.61 + 1.17i)47-s + (−0.809 + 0.587i)48-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 + 0.951i)9-s − 12-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)20-s + 2·23-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (0.618 + 1.90i)31-s + (0.809 + 0.587i)36-s − 45-s + (1.61 + 1.17i)47-s + (−0.809 + 0.587i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.040127596\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.040127596\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - 2T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 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
−9.673194106687871550114171263981, −8.471271661840438494177204270775, −7.44163297650159238059961214938, −6.92250791270006698585727367983, −6.46240036686568621249048227451, −5.52970067268537253988029322906, −4.77869775618248663827716512182, −3.26711293045039118533479787536, −2.38992379716021432069759903252, −1.16989773128452195689913651419,
1.10845664957994227174785695838, 2.68851887938753586690287345124, 3.82768895340007164308622845790, 4.52182196098430655038893368790, 5.45847000480893646433914717494, 6.20629981546640164414514760135, 7.15444785235088647263094210844, 7.82150359324034495852420559027, 8.862271876187065228187162054330, 9.360139616998856294131972830084