| L(s) = 1 | + 3-s + 4·7-s − 2·9-s − 2·13-s − 4·16-s + 4·21-s − 5·27-s + 4·31-s − 16·37-s − 2·39-s + 2·43-s − 4·48-s − 2·49-s − 26·61-s − 8·63-s + 4·67-s + 12·73-s + 30·79-s + 81-s − 8·91-s + 4·93-s − 16·97-s + 22·103-s + 20·109-s − 16·111-s − 16·112-s + 4·117-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s − 0.554·13-s − 16-s + 0.872·21-s − 0.962·27-s + 0.718·31-s − 2.63·37-s − 0.320·39-s + 0.304·43-s − 0.577·48-s − 2/7·49-s − 3.32·61-s − 1.00·63-s + 0.488·67-s + 1.40·73-s + 3.37·79-s + 1/9·81-s − 0.838·91-s + 0.414·93-s − 1.62·97-s + 2.16·103-s + 1.91·109-s − 1.51·111-s − 1.51·112-s + 0.369·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−7.79278253980066871819257147589, −7.72070611477353409994103950518, −7.23778612686470899965206676865, −6.42739878412204514749440358407, −6.41493871731547586436271874024, −5.50737120413184810439847634712, −5.12771377561824053934514670745, −4.72257746414996013390202582736, −4.43338064598993885453060354142, −3.44740587880108288114904722723, −3.32039163045380863323627428423, −2.21415072325817682075527365861, −2.19303688852089224841075154578, −1.33017423399357553642958647309, 0,
1.33017423399357553642958647309, 2.19303688852089224841075154578, 2.21415072325817682075527365861, 3.32039163045380863323627428423, 3.44740587880108288114904722723, 4.43338064598993885453060354142, 4.72257746414996013390202582736, 5.12771377561824053934514670745, 5.50737120413184810439847634712, 6.41493871731547586436271874024, 6.42739878412204514749440358407, 7.23778612686470899965206676865, 7.72070611477353409994103950518, 7.79278253980066871819257147589
