| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.450877624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450877624\) |
| \(L(1)\) |
|
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
−10.33095421162463641514248571015, −9.850256852350633614505772997613, −9.759502643246273420214755907651, −8.550831659664230657872268834606, −8.550525623795299051862865294229, −8.125603444344571218299691627371, −7.57998966343385209913127258383, −7.52695876646109655970032872526, −6.79740997810864848951290662490, −6.29497973143128202937679239261, −6.24306152334201981136965677937, −5.66669141481264212390760281664, −5.04378466490444627673788989768, −4.94468631215194816503747846857, −3.99154359024583216622674382151, −3.08710890079369849769592324563, −3.01317513250868531146554874928, −2.48821146981452051842978688500, −2.11137506925356803682352307925, −1.11280477851658617413417186044,
1.11280477851658617413417186044, 2.11137506925356803682352307925, 2.48821146981452051842978688500, 3.01317513250868531146554874928, 3.08710890079369849769592324563, 3.99154359024583216622674382151, 4.94468631215194816503747846857, 5.04378466490444627673788989768, 5.66669141481264212390760281664, 6.24306152334201981136965677937, 6.29497973143128202937679239261, 6.79740997810864848951290662490, 7.52695876646109655970032872526, 7.57998966343385209913127258383, 8.125603444344571218299691627371, 8.550525623795299051862865294229, 8.550831659664230657872268834606, 9.759502643246273420214755907651, 9.850256852350633614505772997613, 10.33095421162463641514248571015
