| L(s) = 1 | − 9-s − 4·13-s − 2·17-s − 8·19-s + 6·25-s − 16·43-s + 10·49-s − 4·53-s − 8·59-s − 24·67-s + 81-s − 24·83-s − 12·89-s − 12·101-s + 4·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 14·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.10·13-s − 0.485·17-s − 1.83·19-s + 6/5·25-s − 2.43·43-s + 10/7·49-s − 0.549·53-s − 1.04·59-s − 2.93·67-s + 1/9·81-s − 2.63·83-s − 1.27·89-s − 1.19·101-s + 0.369·117-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10653696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10653696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.428602581108460320200220546494, −8.308214659363619700251168385294, −7.62681910073560329533321596098, −7.34927671293210359354178302810, −6.91196014220385073134643908580, −6.69344832447220363833996743583, −6.16862100960228138437264922416, −5.93011284460378139820141700956, −5.36373660397584487843703308909, −4.95037755790314960174178662396, −4.50696747772029112664654524817, −4.39562714406982276701382240793, −3.75195544275812733848703153186, −3.20349456927637199665081096295, −2.63384933237430905026090025816, −2.53177668592740285158076870180, −1.71970836998179909189495326566, −1.33236530530530222750183972939, 0, 0,
1.33236530530530222750183972939, 1.71970836998179909189495326566, 2.53177668592740285158076870180, 2.63384933237430905026090025816, 3.20349456927637199665081096295, 3.75195544275812733848703153186, 4.39562714406982276701382240793, 4.50696747772029112664654524817, 4.95037755790314960174178662396, 5.36373660397584487843703308909, 5.93011284460378139820141700956, 6.16862100960228138437264922416, 6.69344832447220363833996743583, 6.91196014220385073134643908580, 7.34927671293210359354178302810, 7.62681910073560329533321596098, 8.308214659363619700251168385294, 8.428602581108460320200220546494
