| L(s) = 1 | − 3-s − 4·7-s + 9-s + 3·11-s − 13-s + 6·17-s − 6·19-s + 4·21-s − 7·23-s − 27-s − 4·29-s + 11·31-s − 3·33-s − 8·37-s + 39-s + 3·41-s − 43-s − 47-s + 9·49-s − 6·51-s − 11·53-s + 6·57-s − 7·59-s − 8·61-s − 4·63-s − 4·67-s + 7·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 1.45·17-s − 1.37·19-s + 0.872·21-s − 1.45·23-s − 0.192·27-s − 0.742·29-s + 1.97·31-s − 0.522·33-s − 1.31·37-s + 0.160·39-s + 0.468·41-s − 0.152·43-s − 0.145·47-s + 9/7·49-s − 0.840·51-s − 1.51·53-s + 0.794·57-s − 0.911·59-s − 1.02·61-s − 0.503·63-s − 0.488·67-s + 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.15234478105514, −12.58938726735133, −12.34707151402874, −12.03302716720847, −11.51795990803688, −10.87230693966384, −10.32444340880816, −9.932624859416916, −9.743791251591679, −9.107042323222094, −8.596046755268370, −7.984236091907591, −7.507884434809641, −6.862735828784017, −6.457341731572849, −5.983813696032301, −5.860589958025146, −4.960921109331999, −4.322066177229194, −4.002468068514402, −3.195613226145759, −3.032479828785706, −1.996987001925053, −1.513356576979102, −0.6058873113211206, 0,
0.6058873113211206, 1.513356576979102, 1.996987001925053, 3.032479828785706, 3.195613226145759, 4.002468068514402, 4.322066177229194, 4.960921109331999, 5.860589958025146, 5.983813696032301, 6.457341731572849, 6.862735828784017, 7.507884434809641, 7.984236091907591, 8.596046755268370, 9.107042323222094, 9.743791251591679, 9.932624859416916, 10.32444340880816, 10.87230693966384, 11.51795990803688, 12.03302716720847, 12.34707151402874, 12.58938726735133, 13.15234478105514
