L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 4·7-s − 3·8-s − 2·9-s + 10-s + 6·11-s − 12-s − 6·13-s + 4·14-s + 15-s − 16-s + 7·17-s − 2·18-s − 6·19-s − 20-s + 4·21-s + 6·22-s + 3·23-s − 3·24-s + 25-s − 6·26-s − 5·27-s − 4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 1.06·8-s − 2/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s − 1.66·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s + 1.69·17-s − 0.471·18-s − 1.37·19-s − 0.223·20-s + 0.872·21-s + 1.27·22-s + 0.625·23-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.962·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.060548903\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.060548903\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.999954527332220181611149642685, −7.11692004703838727933686991768, −6.29474430295674671658667057635, −5.57889630298583499928028399216, −4.84471189547055451437510864104, −4.47697106852654160159340325026, −3.56925502585548354659876220671, −2.78108337525117715361891063597, −1.93699454646818929363376666616, −0.910078847430348026404658765199,
0.910078847430348026404658765199, 1.93699454646818929363376666616, 2.78108337525117715361891063597, 3.56925502585548354659876220671, 4.47697106852654160159340325026, 4.84471189547055451437510864104, 5.57889630298583499928028399216, 6.29474430295674671658667057635, 7.11692004703838727933686991768, 7.999954527332220181611149642685