L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s + 7-s − 3·8-s + 9-s − 10-s − 3·11-s − 12-s + 7·13-s + 14-s − 15-s − 16-s + 18-s + 8·19-s + 20-s + 21-s − 3·22-s + 4·23-s − 3·24-s + 25-s + 7·26-s + 27-s − 28-s + 2·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.288·12-s + 1.94·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s − 0.639·22-s + 0.834·23-s − 0.612·24-s + 1/5·25-s + 1.37·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.965060215\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.965060215\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.08775017960124, −14.49418668209324, −13.97132918840065, −13.53493056319301, −13.18447751358834, −12.70130916920717, −11.99297841067912, −11.45812745706353, −10.95335677265559, −10.34814825795823, −9.620777482829189, −9.027843443885381, −8.551689988535591, −8.188901724450319, −7.369715868626766, −7.024951046395269, −5.848320434607759, −5.619013588641349, −4.993060353811862, −4.097619807624086, −3.870937306472872, −3.044854796044160, −2.676705372739451, −1.353561790182197, −0.7390257258645114,
0.7390257258645114, 1.353561790182197, 2.676705372739451, 3.044854796044160, 3.870937306472872, 4.097619807624086, 4.993060353811862, 5.619013588641349, 5.848320434607759, 7.024951046395269, 7.369715868626766, 8.188901724450319, 8.551689988535591, 9.027843443885381, 9.620777482829189, 10.34814825795823, 10.95335677265559, 11.45812745706353, 11.99297841067912, 12.70130916920717, 13.18447751358834, 13.53493056319301, 13.97132918840065, 14.49418668209324, 15.08775017960124