| L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 2·11-s − 5·13-s − 3·17-s − 5·19-s + 4·21-s + 2·23-s + 4·27-s − 7·29-s − 3·31-s + 4·33-s − 8·37-s + 10·39-s + 5·41-s + 9·43-s + 9·47-s − 3·49-s + 6·51-s + 8·53-s + 10·57-s + 6·59-s + 61-s − 2·63-s + 7·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 1.38·13-s − 0.727·17-s − 1.14·19-s + 0.872·21-s + 0.417·23-s + 0.769·27-s − 1.29·29-s − 0.538·31-s + 0.696·33-s − 1.31·37-s + 1.60·39-s + 0.780·41-s + 1.37·43-s + 1.31·47-s − 3/7·49-s + 0.840·51-s + 1.09·53-s + 1.32·57-s + 0.781·59-s + 0.128·61-s − 0.251·63-s + 0.855·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.78980394781277, −15.20131391599518, −14.67361712255468, −14.12543603821124, −13.23865044586118, −12.87281617044480, −12.45653389552358, −11.94702521052370, −11.25052008622133, −10.79971095681821, −10.34601126089610, −9.799951972839635, −9.001593825013081, −8.711807254610779, −7.597461309414115, −7.204408168631357, −6.669662011959429, −5.910218084504156, −5.565428328019172, −4.861652940168340, −4.304623437732543, −3.479493410872375, −2.524874282937565, −2.078135708290209, −0.6281503383325155, 0,
0.6281503383325155, 2.078135708290209, 2.524874282937565, 3.479493410872375, 4.304623437732543, 4.861652940168340, 5.565428328019172, 5.910218084504156, 6.669662011959429, 7.204408168631357, 7.597461309414115, 8.711807254610779, 9.001593825013081, 9.799951972839635, 10.34601126089610, 10.79971095681821, 11.25052008622133, 11.94702521052370, 12.45653389552358, 12.87281617044480, 13.23865044586118, 14.12543603821124, 14.67361712255468, 15.20131391599518, 15.78980394781277
