L(s) = 1 | − 2·3-s − 2·5-s − 5·7-s + 9-s − 6·11-s − 6·13-s + 4·15-s − 3·17-s − 8·19-s + 10·21-s − 2·23-s − 25-s + 4·27-s + 2·29-s − 9·31-s + 12·33-s + 10·35-s − 4·37-s + 12·39-s − 6·41-s − 12·43-s − 2·45-s + 3·47-s + 18·49-s + 6·51-s + 3·53-s + 12·55-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s − 1.88·7-s + 1/3·9-s − 1.80·11-s − 1.66·13-s + 1.03·15-s − 0.727·17-s − 1.83·19-s + 2.18·21-s − 0.417·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s − 1.61·31-s + 2.08·33-s + 1.69·35-s − 0.657·37-s + 1.92·39-s − 0.937·41-s − 1.82·43-s − 0.298·45-s + 0.437·47-s + 18/7·49-s + 0.840·51-s + 0.412·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 6571 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−16.28783902585403, −15.63085519753034, −15.17974471448100, −14.85108039787864, −13.76617892649391, −13.14633390513072, −12.69938646170797, −12.51391685969301, −11.81863384711516, −11.36680393471989, −10.48808977766369, −10.28181409061573, −9.950524865024282, −8.898408694194140, −8.504891867067960, −7.620618398444810, −7.061695267555376, −6.730112756797060, −5.942049684660046, −5.482543017149340, −4.783586214425482, −4.223301619979378, −3.366145361083828, −2.712816972926759, −2.083287004050114, 0, 0, 0,
2.083287004050114, 2.712816972926759, 3.366145361083828, 4.223301619979378, 4.783586214425482, 5.482543017149340, 5.942049684660046, 6.730112756797060, 7.061695267555376, 7.620618398444810, 8.504891867067960, 8.898408694194140, 9.950524865024282, 10.28181409061573, 10.48808977766369, 11.36680393471989, 11.81863384711516, 12.51391685969301, 12.69938646170797, 13.14633390513072, 13.76617892649391, 14.85108039787864, 15.17974471448100, 15.63085519753034, 16.28783902585403