L(s) = 1 | − 5-s + 4·7-s − 2·9-s + 2·13-s + 25-s + 12·29-s − 4·35-s + 4·37-s + 2·45-s − 12·47-s − 2·49-s + 4·61-s − 8·63-s − 2·65-s + 4·67-s + 4·73-s + 16·79-s − 5·81-s + 12·83-s + 8·91-s + 4·97-s + 12·101-s − 4·117-s − 22·121-s − 125-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s − 2/3·9-s + 0.554·13-s + 1/5·25-s + 2.22·29-s − 0.676·35-s + 0.657·37-s + 0.298·45-s − 1.75·47-s − 2/7·49-s + 0.512·61-s − 1.00·63-s − 0.248·65-s + 0.488·67-s + 0.468·73-s + 1.80·79-s − 5/9·81-s + 1.31·83-s + 0.838·91-s + 0.406·97-s + 1.19·101-s − 0.369·117-s − 2·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.978873707\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.978873707\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( 1 + T \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.552217550781204179646223792184, −8.272247941931688318313747530511, −7.935122978007669521864351036685, −7.64058951798313836751383325584, −6.67779207539369308128578318805, −6.57891116465648258947670054106, −5.92025328033368364574944400039, −5.25425303036343223544000159628, −4.78130792717525308450176413839, −4.57800936747163757853558845093, −3.73611749392351924270213176061, −3.19605148689446138736614927407, −2.48783872012348165896827254358, −1.70743027705458628297492265631, −0.853693834944140891042010218471,
0.853693834944140891042010218471, 1.70743027705458628297492265631, 2.48783872012348165896827254358, 3.19605148689446138736614927407, 3.73611749392351924270213176061, 4.57800936747163757853558845093, 4.78130792717525308450176413839, 5.25425303036343223544000159628, 5.92025328033368364574944400039, 6.57891116465648258947670054106, 6.67779207539369308128578318805, 7.64058951798313836751383325584, 7.935122978007669521864351036685, 8.272247941931688318313747530511, 8.552217550781204179646223792184