L(s) = 1 | − 7-s − 4·11-s − 6·17-s + 4·19-s + 4·23-s − 5·25-s + 2·29-s − 8·31-s − 6·37-s + 6·41-s + 4·43-s + 6·47-s + 49-s + 6·53-s − 12·59-s + 8·61-s − 6·67-s − 12·71-s + 2·73-s + 4·77-s − 12·79-s − 10·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.20·11-s − 1.45·17-s + 0.917·19-s + 0.834·23-s − 25-s + 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 1.02·61-s − 0.733·67-s − 1.42·71-s + 0.234·73-s + 0.455·77-s − 1.35·79-s − 1.05·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7499523606\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7499523606\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−13.75913370707442, −13.41828290690935, −13.02398355997854, −12.51521051417218, −12.00457746768632, −11.35440058652617, −10.91233567235375, −10.53664177475750, −9.952495623323725, −9.377150589547651, −8.903596566860741, −8.516758216972748, −7.679387574286379, −7.308914645542721, −6.969402740051025, −6.070987944343457, −5.700532897504553, −5.148085279920903, −4.520687419907008, −3.948262398405824, −3.235654538250052, −2.648736592375707, −2.138509841515715, −1.286429373385798, −0.2788778448218502,
0.2788778448218502, 1.286429373385798, 2.138509841515715, 2.648736592375707, 3.235654538250052, 3.948262398405824, 4.520687419907008, 5.148085279920903, 5.700532897504553, 6.070987944343457, 6.969402740051025, 7.308914645542721, 7.679387574286379, 8.516758216972748, 8.903596566860741, 9.377150589547651, 9.952495623323725, 10.53664177475750, 10.91233567235375, 11.35440058652617, 12.00457746768632, 12.51521051417218, 13.02398355997854, 13.41828290690935, 13.75913370707442