L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s − 15-s + 16-s − 3·17-s + 18-s − 19-s − 20-s + 21-s + 22-s + 3·23-s + 24-s − 4·25-s − 26-s + 27-s + 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 + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−15.41774792116834, −15.00397338283526, −14.63337523832411, −14.11507898612722, −13.47761463615517, −12.94934844472169, −12.68140116941841, −11.70688893647750, −11.45939504478501, −11.03631433562606, −10.03625775621487, −9.811087453551154, −8.817287715094643, −8.536232454775946, −7.663852861567746, −7.437885720319620, −6.543255027867653, −6.183723339719197, −5.190579888437125, −4.667750370866134, −4.144880570804156, −3.446003408066807, −2.832755153440247, −2.028714624674828, −1.355629463398903, 0,
1.355629463398903, 2.028714624674828, 2.832755153440247, 3.446003408066807, 4.144880570804156, 4.667750370866134, 5.190579888437125, 6.183723339719197, 6.543255027867653, 7.437885720319620, 7.663852861567746, 8.536232454775946, 8.817287715094643, 9.811087453551154, 10.03625775621487, 11.03631433562606, 11.45939504478501, 11.70688893647750, 12.68140116941841, 12.94934844472169, 13.47761463615517, 14.11507898612722, 14.63337523832411, 15.00397338283526, 15.41774792116834