| L(s) = 1 | − 3·3-s − 3·4-s − 5·7-s + 6·9-s + 9·12-s − 9·13-s + 5·16-s − 10·19-s + 15·21-s − 9·25-s − 9·27-s + 15·28-s − 13·31-s − 18·36-s − 8·37-s + 27·39-s − 13·43-s − 15·48-s + 14·49-s + 27·52-s + 30·57-s − 12·61-s − 30·63-s − 3·64-s + 2·67-s + 7·73-s + 27·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 3/2·4-s − 1.88·7-s + 2·9-s + 2.59·12-s − 2.49·13-s + 5/4·16-s − 2.29·19-s + 3.27·21-s − 9/5·25-s − 1.73·27-s + 2.83·28-s − 2.33·31-s − 3·36-s − 1.31·37-s + 4.32·39-s − 1.98·43-s − 2.16·48-s + 2·49-s + 3.74·52-s + 3.97·57-s − 1.53·61-s − 3.77·63-s − 3/8·64-s + 0.244·67-s + 0.819·73-s + 3.11·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1297737 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1297737 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 20599 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 160 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.10772135264045952946056295298, −6.80778758743189815159114627469, −6.49630676256752051258118064004, −5.83653724390730432483627791083, −5.59586287405651631704175165840, −5.13112254588624048810966011994, −4.66884555453797505385033799712, −4.29606892864498012105401518636, −3.77609676899423195183939508662, −3.35015228904337405476899581372, −2.35483510466624075402768449386, −1.76492895722032538972584174092, 0, 0, 0,
1.76492895722032538972584174092, 2.35483510466624075402768449386, 3.35015228904337405476899581372, 3.77609676899423195183939508662, 4.29606892864498012105401518636, 4.66884555453797505385033799712, 5.13112254588624048810966011994, 5.59586287405651631704175165840, 5.83653724390730432483627791083, 6.49630676256752051258118064004, 6.80778758743189815159114627469, 7.10772135264045952946056295298
