| L(s) = 1 | + 2·5-s − 5·9-s + 11-s + 8·13-s − 4·16-s − 4·17-s − 7·25-s + 7·31-s − 12·43-s − 10·45-s + 16·47-s − 10·49-s + 2·55-s + 10·59-s + 24·61-s + 16·65-s − 14·67-s − 6·71-s + 8·73-s − 20·79-s − 8·80-s + 16·81-s − 12·83-s − 8·85-s − 14·97-s − 5·99-s − 32·103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 5/3·9-s + 0.301·11-s + 2.21·13-s − 16-s − 0.970·17-s − 7/5·25-s + 1.25·31-s − 1.82·43-s − 1.49·45-s + 2.33·47-s − 1.42·49-s + 0.269·55-s + 1.30·59-s + 3.07·61-s + 1.98·65-s − 1.71·67-s − 0.712·71-s + 0.936·73-s − 2.25·79-s − 0.894·80-s + 16/9·81-s − 1.31·83-s − 0.867·85-s − 1.42·97-s − 0.502·99-s − 3.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1279091 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1279091 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$ | \( 1 - T \) |
| 31 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−8.161876221686647141725563527465, −7.10826142797358908090370373880, −6.82697224947399950023698351410, −6.36261389471308870138602900888, −6.00363755168830900564762744465, −5.61864640510783414285306575446, −5.41935991651992802218472234441, −4.51842699512191534356663710538, −4.11715899745388408852378587541, −3.63170186460888703246647999593, −2.97562431456794399623394713691, −2.44672211519329826292315815599, −1.92484566977028663873402211272, −1.17041086963532389621044182318, 0,
1.17041086963532389621044182318, 1.92484566977028663873402211272, 2.44672211519329826292315815599, 2.97562431456794399623394713691, 3.63170186460888703246647999593, 4.11715899745388408852378587541, 4.51842699512191534356663710538, 5.41935991651992802218472234441, 5.61864640510783414285306575446, 6.00363755168830900564762744465, 6.36261389471308870138602900888, 6.82697224947399950023698351410, 7.10826142797358908090370373880, 8.161876221686647141725563527465
