L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 5·11-s − 12-s + 13-s + 14-s + 16-s − 3·17-s − 18-s + 21-s + 5·22-s − 23-s + 24-s − 26-s − 27-s − 28-s + 29-s + 2·31-s − 32-s + 5·33-s + 3·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 0.208·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.185·29-s + 0.359·31-s − 0.176·32-s + 0.870·33-s + 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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
−12.60387315105168, −12.14517202666243, −11.86372058649876, −11.01801503426687, −10.93462822559450, −10.55222404253607, −9.948103173243336, −9.685123142865427, −9.124637192323382, −8.568067583658594, −8.124791352674542, −7.765964352640649, −7.216875119380225, −6.697582457410390, −6.340742210971913, −5.782139286620926, −5.368721539369961, −4.732029931206766, −4.399191387751227, −3.586132799979709, −2.898261647799114, −2.723465910154997, −1.759550668038256, −1.480932309985668, −0.4350303925098992, 0,
0.4350303925098992, 1.480932309985668, 1.759550668038256, 2.723465910154997, 2.898261647799114, 3.586132799979709, 4.399191387751227, 4.732029931206766, 5.368721539369961, 5.782139286620926, 6.340742210971913, 6.697582457410390, 7.216875119380225, 7.765964352640649, 8.124791352674542, 8.568067583658594, 9.124637192323382, 9.685123142865427, 9.948103173243336, 10.55222404253607, 10.93462822559450, 11.01801503426687, 11.86372058649876, 12.14517202666243, 12.60387315105168