| L(s) = 1 | + 2·4-s − 9-s + 2·11-s + 3·16-s − 25-s − 4·29-s − 2·36-s + 4·44-s − 49-s + 4·64-s + 81-s − 2·99-s − 2·100-s − 8·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2·4-s − 9-s + 2·11-s + 3·16-s − 25-s − 4·29-s − 2·36-s + 4·44-s − 49-s + 4·64-s + 81-s − 2·99-s − 2·100-s − 8·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.713336548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713336548\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−10.23467578696084954665378544818, −9.625876145539839012583007961089, −9.527384056467353278988998414419, −9.028362675597971356073101626326, −8.581791916309551491610839674765, −7.894339687283806698440031033957, −7.75499352573368707785935945192, −7.25569971866949852766409564524, −6.78069025635358036221574517904, −6.58318811242114638801919333938, −5.88377385981323561375903562525, −5.78551983750898397389696896877, −5.47183495480473603198169839811, −4.51667698211988893014933190663, −3.72476401610692384551673508445, −3.60117938565520584711105417193, −3.11752923669926194729028171336, −2.19347572953911401320311710685, −1.94184654146782798660644617949, −1.35306713622553753215455472887,
1.35306713622553753215455472887, 1.94184654146782798660644617949, 2.19347572953911401320311710685, 3.11752923669926194729028171336, 3.60117938565520584711105417193, 3.72476401610692384551673508445, 4.51667698211988893014933190663, 5.47183495480473603198169839811, 5.78551983750898397389696896877, 5.88377385981323561375903562525, 6.58318811242114638801919333938, 6.78069025635358036221574517904, 7.25569971866949852766409564524, 7.75499352573368707785935945192, 7.894339687283806698440031033957, 8.581791916309551491610839674765, 9.028362675597971356073101626326, 9.527384056467353278988998414419, 9.625876145539839012583007961089, 10.23467578696084954665378544818
