| L(s) = 1 | − 2-s + 2·3-s + 4-s + 3·5-s − 2·6-s + 3·7-s − 3·8-s + 3·9-s − 3·10-s + 4·11-s + 2·12-s − 3·14-s + 6·15-s + 16-s + 17-s − 3·18-s − 6·19-s + 3·20-s + 6·21-s − 4·22-s + 4·23-s − 6·24-s + 25-s + 4·27-s + 3·28-s + 29-s − 6·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 1.34·5-s − 0.816·6-s + 1.13·7-s − 1.06·8-s + 9-s − 0.948·10-s + 1.20·11-s + 0.577·12-s − 0.801·14-s + 1.54·15-s + 1/4·16-s + 0.242·17-s − 0.707·18-s − 1.37·19-s + 0.670·20-s + 1.30·21-s − 0.852·22-s + 0.834·23-s − 1.22·24-s + 1/5·25-s + 0.769·27-s + 0.566·28-s + 0.185·29-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.902178037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.902178037\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 88 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 98 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 169 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 136 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 165 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 15 T + 210 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 242 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 13 T + 232 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.86746015591716693988632929975, −10.68832318616732055991105685269, −9.937095145303307375204410934467, −9.846403248930172989474048516157, −9.131413003359716976605778254528, −8.992562004097801047001551216995, −8.531346012746000623099240922277, −8.327150364814828599349628820907, −7.64231171039695604004499274615, −6.95723307936731005989370733517, −6.71600125421775928105928531431, −6.25886535335637483940327810364, −5.48589071181208357151227696095, −5.18038228423165673520942204067, −4.22577094883249025782498773303, −3.86824313279929653220187459391, −2.99142665580051132207938482349, −2.27452445155264935855313400762, −1.93481065374349873308465218165, −1.21501910271491189756816325506,
1.21501910271491189756816325506, 1.93481065374349873308465218165, 2.27452445155264935855313400762, 2.99142665580051132207938482349, 3.86824313279929653220187459391, 4.22577094883249025782498773303, 5.18038228423165673520942204067, 5.48589071181208357151227696095, 6.25886535335637483940327810364, 6.71600125421775928105928531431, 6.95723307936731005989370733517, 7.64231171039695604004499274615, 8.327150364814828599349628820907, 8.531346012746000623099240922277, 8.992562004097801047001551216995, 9.131413003359716976605778254528, 9.846403248930172989474048516157, 9.937095145303307375204410934467, 10.68832318616732055991105685269, 10.86746015591716693988632929975
