| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 3·7-s − 2·10-s − 11-s − 13-s + 6·14-s − 4·16-s − 2·19-s + 2·20-s + 2·22-s − 3·23-s + 25-s + 2·26-s − 6·28-s − 2·29-s + 6·31-s + 8·32-s − 3·35-s − 11·37-s + 4·38-s − 5·41-s + 4·43-s − 2·44-s + 6·46-s + 10·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 1.13·7-s − 0.632·10-s − 0.301·11-s − 0.277·13-s + 1.60·14-s − 16-s − 0.458·19-s + 0.447·20-s + 0.426·22-s − 0.625·23-s + 1/5·25-s + 0.392·26-s − 1.13·28-s − 0.371·29-s + 1.07·31-s + 1.41·32-s − 0.507·35-s − 1.80·37-s + 0.648·38-s − 0.780·41-s + 0.609·43-s − 0.301·44-s + 0.884·46-s + 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.55417159764438, −12.93802748279101, −12.34484834497242, −12.16006200756759, −11.40094659390147, −10.74516978690745, −10.48803120676034, −10.06788899702383, −9.611244356660486, −9.230921883911681, −8.766350543210205, −8.331027252703295, −7.629284949560462, −7.406615500495284, −6.601071344588262, −6.414253270072652, −5.835473142323705, −5.087972806031363, −4.576375145547657, −3.864310088183347, −3.175342256017222, −2.647830596245179, −1.974056809841830, −1.490503176197172, −0.5625334335289480, 0,
0.5625334335289480, 1.490503176197172, 1.974056809841830, 2.647830596245179, 3.175342256017222, 3.864310088183347, 4.576375145547657, 5.087972806031363, 5.835473142323705, 6.414253270072652, 6.601071344588262, 7.406615500495284, 7.629284949560462, 8.331027252703295, 8.766350543210205, 9.230921883911681, 9.611244356660486, 10.06788899702383, 10.48803120676034, 10.74516978690745, 11.40094659390147, 12.16006200756759, 12.34484834497242, 12.93802748279101, 13.55417159764438
