L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 4·10-s − 11-s − 13-s − 4·16-s + 17-s + 2·19-s + 4·20-s + 2·22-s − 4·23-s − 25-s + 2·26-s + 5·31-s + 8·32-s − 2·34-s + 5·37-s − 4·38-s − 9·41-s − 13·43-s − 2·44-s + 8·46-s + 6·47-s + 2·50-s − 2·52-s + 3·53-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 1.26·10-s − 0.301·11-s − 0.277·13-s − 16-s + 0.242·17-s + 0.458·19-s + 0.894·20-s + 0.426·22-s − 0.834·23-s − 1/5·25-s + 0.392·26-s + 0.898·31-s + 1.41·32-s − 0.342·34-s + 0.821·37-s − 0.648·38-s − 1.40·41-s − 1.98·43-s − 0.301·44-s + 1.17·46-s + 0.875·47-s + 0.282·50-s − 0.277·52-s + 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8825631361\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8825631361\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.71806469482197, −13.33391313352721, −12.95748069395766, −12.04209738303854, −11.70416950663647, −11.30791832848230, −10.42129715430170, −10.14523117376456, −9.910615099256332, −9.445481967091962, −8.787378518525144, −8.344053101065590, −7.955844433106420, −7.361180046737955, −6.758578398091058, −6.418178865581556, −5.523548844138878, −5.321119643113563, −4.459700862358044, −3.873068441374899, −2.931644799508766, −2.410570238504415, −1.764401355966888, −1.252092368701913, −0.3888309355545605,
0.3888309355545605, 1.252092368701913, 1.764401355966888, 2.410570238504415, 2.931644799508766, 3.873068441374899, 4.459700862358044, 5.321119643113563, 5.523548844138878, 6.418178865581556, 6.758578398091058, 7.361180046737955, 7.955844433106420, 8.344053101065590, 8.787378518525144, 9.445481967091962, 9.910615099256332, 10.14523117376456, 10.42129715430170, 11.30791832848230, 11.70416950663647, 12.04209738303854, 12.95748069395766, 13.33391313352721, 13.71806469482197