| L(s) = 1 | − 2·2-s − 4-s + 8·8-s + 4·13-s − 7·16-s − 2·17-s − 8·19-s + 10·25-s − 8·26-s − 14·32-s + 4·34-s + 16·38-s − 8·43-s + 16·47-s − 2·49-s − 20·50-s − 4·52-s − 12·53-s + 24·59-s + 35·64-s + 24·67-s + 2·68-s + 8·76-s − 24·83-s + 16·86-s + 20·89-s − 32·94-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 2.82·8-s + 1.10·13-s − 7/4·16-s − 0.485·17-s − 1.83·19-s + 2·25-s − 1.56·26-s − 2.47·32-s + 0.685·34-s + 2.59·38-s − 1.21·43-s + 2.33·47-s − 2/7·49-s − 2.82·50-s − 0.554·52-s − 1.64·53-s + 3.12·59-s + 35/8·64-s + 2.93·67-s + 0.242·68-s + 0.917·76-s − 2.63·83-s + 1.72·86-s + 2.11·89-s − 3.30·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4140331733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4140331733\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−13.20562517273108978229664066351, −12.79183441380822445737485066841, −12.47673269856562940104771077002, −11.34622133274029039988953571775, −10.93703098429816637646947671019, −10.67019276252676296635009348463, −10.02120320190170304188941308070, −9.700904361232117750200167263973, −8.803122490443163463083083304378, −8.614666728887835615841036671452, −8.525512840279562471351607821652, −7.73703964893348709714163440024, −6.94874426170867693671852723273, −6.55314321322967584637318361014, −5.54109626339049705766674438727, −4.82491262168358477569632493973, −4.26184654181768534650831433465, −3.60800496623101173067285949740, −2.10376560199305166100303049316, −0.864208652357659300161169448728,
0.864208652357659300161169448728, 2.10376560199305166100303049316, 3.60800496623101173067285949740, 4.26184654181768534650831433465, 4.82491262168358477569632493973, 5.54109626339049705766674438727, 6.55314321322967584637318361014, 6.94874426170867693671852723273, 7.73703964893348709714163440024, 8.525512840279562471351607821652, 8.614666728887835615841036671452, 8.803122490443163463083083304378, 9.700904361232117750200167263973, 10.02120320190170304188941308070, 10.67019276252676296635009348463, 10.93703098429816637646947671019, 11.34622133274029039988953571775, 12.47673269856562940104771077002, 12.79183441380822445737485066841, 13.20562517273108978229664066351
