L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 8-s + 9-s + 2·10-s + 12-s + 2·15-s + 16-s + 18-s + 10·19-s + 2·20-s + 9·23-s + 24-s + 2·25-s + 27-s + 2·29-s + 2·30-s + 32-s + 36-s + 10·38-s + 2·40-s − 17·43-s + 2·45-s + 9·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.235·18-s + 2.29·19-s + 0.447·20-s + 1.87·23-s + 0.204·24-s + 2/5·25-s + 0.192·27-s + 0.371·29-s + 0.365·30-s + 0.176·32-s + 1/6·36-s + 1.62·38-s + 0.316·40-s − 2.59·43-s + 0.298·45-s + 1.32·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.171965653\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.171965653\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.504781415709807615237586525525, −8.055662029429704928057425386675, −7.45518187231234921921730817212, −7.09786251393495455961485465523, −6.75954741136755292216897899642, −6.16977726681192307014958542351, −5.60561577994433280389025704263, −5.15284014000142106284220981894, −4.90631824974596969246171556657, −4.22138532680013923779013007571, −3.39614605186010541812941081516, −3.05277416540107206132681299710, −2.68631632072872926966463720836, −1.65694028363211593343744867692, −1.20918252580375677757506356202,
1.20918252580375677757506356202, 1.65694028363211593343744867692, 2.68631632072872926966463720836, 3.05277416540107206132681299710, 3.39614605186010541812941081516, 4.22138532680013923779013007571, 4.90631824974596969246171556657, 5.15284014000142106284220981894, 5.60561577994433280389025704263, 6.16977726681192307014958542351, 6.75954741136755292216897899642, 7.09786251393495455961485465523, 7.45518187231234921921730817212, 8.055662029429704928057425386675, 8.504781415709807615237586525525