L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 4·13-s − 14-s + 16-s − 17-s + 19-s + 20-s + 25-s + 4·26-s − 28-s + 2·29-s + 6·31-s + 32-s − 34-s − 35-s − 3·37-s + 38-s + 40-s + 8·41-s + 8·43-s − 11·47-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.229·19-s + 0.223·20-s + 1/5·25-s + 0.784·26-s − 0.188·28-s + 0.371·29-s + 1.07·31-s + 0.176·32-s − 0.171·34-s − 0.169·35-s − 0.493·37-s + 0.162·38-s + 0.158·40-s + 1.24·41-s + 1.21·43-s − 1.60·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−12.88779301045218, −12.47137413459328, −11.98795080155767, −11.40992976900259, −11.12800155670472, −10.60979211640201, −10.16333483956534, −9.641714719213658, −9.273736203618599, −8.633288559852617, −8.204435655561924, −7.804433920778623, −6.985300740363407, −6.702758331548959, −6.294162516661909, −5.716347709227733, −5.423238071962771, −4.776863027540085, −4.181745769312416, −3.829579702334181, −3.210796587644418, −2.627896318086282, −2.285993569063383, −1.305887721869496, −1.078244018456141, 0,
1.078244018456141, 1.305887721869496, 2.285993569063383, 2.627896318086282, 3.210796587644418, 3.829579702334181, 4.181745769312416, 4.776863027540085, 5.423238071962771, 5.716347709227733, 6.294162516661909, 6.702758331548959, 6.985300740363407, 7.804433920778623, 8.204435655561924, 8.633288559852617, 9.273736203618599, 9.641714719213658, 10.16333483956534, 10.60979211640201, 11.12800155670472, 11.40992976900259, 11.98795080155767, 12.47137413459328, 12.88779301045218