L(s) = 1 | + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s − 1.61·7-s + (−0.809 + 0.587i)9-s + (1.30 + 0.951i)11-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.809 − 0.587i)20-s + (1.30 + 0.951i)23-s + (0.309 − 0.951i)25-s + (−0.500 − 1.53i)28-s + (1.30 − 0.951i)35-s + (−0.809 − 0.587i)36-s + 0.618·43-s + (−0.499 + 1.53i)44-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s − 1.61·7-s + (−0.809 + 0.587i)9-s + (1.30 + 0.951i)11-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.809 − 0.587i)20-s + (1.30 + 0.951i)23-s + (0.309 − 0.951i)25-s + (−0.500 − 1.53i)28-s + (1.30 − 0.951i)35-s + (−0.809 − 0.587i)36-s + 0.618·43-s + (−0.499 + 1.53i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6601589846\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6601589846\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 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 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.57098765060952982556298482045, −10.82485535285528869782389645456, −9.467055822960684928197943573349, −8.938365329466624269561752040084, −7.50875694392123445162509218525, −7.09503056414150789237536816268, −6.22413521839050327094082296587, −4.51293651688591239879326387800, −3.35540762624933005372024145887, −2.77778598257329082561604699066,
0.872287529666106770604159254580, 3.11529654277132639688758752091, 3.95777207454663036647427554552, 5.58527622991177046775248432152, 6.26518154266451488968075602242, 7.02039870330467664178975716433, 8.650388957596350252990442803839, 9.082053096475251744386313063417, 10.03522390801117895182415332128, 11.03471040038214667143058997547