| L(s) = 1 | − 4-s + 2·7-s − 9-s + 10·11-s + 4·13-s + 16-s − 6·25-s − 2·28-s + 10·29-s + 36-s − 20·37-s + 12·43-s − 10·44-s + 12·47-s − 11·49-s − 4·52-s + 14·53-s − 24·59-s − 2·63-s − 64-s + 20·77-s + 81-s − 28·89-s + 8·91-s + 10·97-s − 10·99-s + 6·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.755·7-s − 1/3·9-s + 3.01·11-s + 1.10·13-s + 1/4·16-s − 6/5·25-s − 0.377·28-s + 1.85·29-s + 1/6·36-s − 3.28·37-s + 1.82·43-s − 1.50·44-s + 1.75·47-s − 1.57·49-s − 0.554·52-s + 1.92·53-s − 3.12·59-s − 0.251·63-s − 1/8·64-s + 2.27·77-s + 1/9·81-s − 2.96·89-s + 0.838·91-s + 1.01·97-s − 1.00·99-s + 3/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.835915627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835915627\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 53 | $C_2$ | \( 1 - 14 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−11.97832324529025757152791005566, −11.51950611919021271440673056792, −10.90281845878098100411503523348, −10.64853138503910323890143443976, −9.976678161283512843816827361792, −9.355928540933491740730661229417, −8.890177834676785330253663348662, −8.841471073307060467201279866998, −8.239704885576410988410465565015, −7.68077436283308438393520759525, −6.74422658776545809366699068131, −6.73866635300919625217018803122, −5.93426590030487536300365765583, −5.57152941841203308210347575606, −4.68040532212025185086542620358, −4.02812088271024247271571331199, −3.90878468206067804936956687278, −3.04304548084901527146420934052, −1.69717907455138203769571317989, −1.20947970709776881780644882711,
1.20947970709776881780644882711, 1.69717907455138203769571317989, 3.04304548084901527146420934052, 3.90878468206067804936956687278, 4.02812088271024247271571331199, 4.68040532212025185086542620358, 5.57152941841203308210347575606, 5.93426590030487536300365765583, 6.73866635300919625217018803122, 6.74422658776545809366699068131, 7.68077436283308438393520759525, 8.239704885576410988410465565015, 8.841471073307060467201279866998, 8.890177834676785330253663348662, 9.355928540933491740730661229417, 9.976678161283512843816827361792, 10.64853138503910323890143443976, 10.90281845878098100411503523348, 11.51950611919021271440673056792, 11.97832324529025757152791005566
