| L(s) = 1 | − 2·3-s − 4-s − 3·9-s + 2·12-s − 6·13-s + 16-s − 4·17-s + 18·23-s + 6·25-s + 14·27-s + 3·36-s + 12·39-s + 8·43-s − 2·48-s − 49-s + 8·51-s + 6·52-s + 28·53-s − 26·61-s − 64-s + 4·68-s − 36·69-s − 12·75-s − 30·79-s − 4·81-s − 18·92-s − 6·100-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 9-s + 0.577·12-s − 1.66·13-s + 1/4·16-s − 0.970·17-s + 3.75·23-s + 6/5·25-s + 2.69·27-s + 1/2·36-s + 1.92·39-s + 1.21·43-s − 0.288·48-s − 1/7·49-s + 1.12·51-s + 0.832·52-s + 3.84·53-s − 3.32·61-s − 1/8·64-s + 0.485·68-s − 4.33·69-s − 1.38·75-s − 3.37·79-s − 4/9·81-s − 1.87·92-s − 3/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5349397619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5349397619\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−12.93119349081355791914340153833, −12.15024766397434821591191326201, −12.06547880623389505221712140311, −11.27816378686614441567988486169, −11.01376410975246372292666022780, −10.61204278326719737762528447609, −10.08743018168375071318857811530, −9.035256301084026429752484754638, −9.007433701956728047461338794138, −8.670775969103889500576613285046, −7.60716817616235682281313644963, −6.94806161405438964235558771450, −6.82936314874885594614201309543, −5.72198202527973584933323052172, −5.49090283545892115720542510907, −4.66655857114829313718055745499, −4.63933661018476044240520595068, −2.96834517469500924454491996503, −2.75166711872665424481880447923, −0.72117616638807463498186169950,
0.72117616638807463498186169950, 2.75166711872665424481880447923, 2.96834517469500924454491996503, 4.63933661018476044240520595068, 4.66655857114829313718055745499, 5.49090283545892115720542510907, 5.72198202527973584933323052172, 6.82936314874885594614201309543, 6.94806161405438964235558771450, 7.60716817616235682281313644963, 8.670775969103889500576613285046, 9.007433701956728047461338794138, 9.035256301084026429752484754638, 10.08743018168375071318857811530, 10.61204278326719737762528447609, 11.01376410975246372292666022780, 11.27816378686614441567988486169, 12.06547880623389505221712140311, 12.15024766397434821591191326201, 12.93119349081355791914340153833
