L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s + 13-s − 15-s + 17-s + 4·19-s + 4·21-s − 8·23-s + 25-s + 27-s − 10·29-s − 4·31-s − 4·35-s + 10·37-s + 39-s − 10·41-s − 4·43-s − 45-s + 8·47-s + 9·49-s + 51-s − 10·53-s + 4·57-s + 4·59-s − 2·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s + 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.676·35-s + 1.64·37-s + 0.160·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.140·51-s − 1.37·53-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.33925939046155, −14.92185684189818, −14.63225787984601, −13.96475814324361, −13.59007199207749, −12.98945685051413, −12.19679927213483, −11.75278498614824, −11.33976779258412, −10.76316045833517, −10.14926649795610, −9.433396190956645, −8.991838349574090, −8.212515731697167, −7.816646368887019, −7.584407579918766, −6.786083101993569, −5.821905927380728, −5.406896704830387, −4.616946537266775, −4.043368257094471, −3.516103161489112, −2.632158044124214, −1.758439708970142, −1.357682047628373, 0,
1.357682047628373, 1.758439708970142, 2.632158044124214, 3.516103161489112, 4.043368257094471, 4.616946537266775, 5.406896704830387, 5.821905927380728, 6.786083101993569, 7.584407579918766, 7.816646368887019, 8.212515731697167, 8.991838349574090, 9.433396190956645, 10.14926649795610, 10.76316045833517, 11.33976779258412, 11.75278498614824, 12.19679927213483, 12.98945685051413, 13.59007199207749, 13.96475814324361, 14.63225787984601, 14.92185684189818, 15.33925939046155