| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s + 15-s + 16-s − 18-s − 8·19-s + 20-s + 6·23-s − 24-s − 2·25-s + 27-s + 7·29-s − 30-s − 32-s + 36-s + 8·38-s − 40-s + 8·43-s + 45-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.223·20-s + 1.25·23-s − 0.204·24-s − 2/5·25-s + 0.192·27-s + 1.29·29-s − 0.182·30-s − 0.176·32-s + 1/6·36-s + 1.29·38-s − 0.158·40-s + 1.21·43-s + 0.149·45-s − 0.884·46-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
| \(L(1)\) |
\(\approx\) |
\(1.758270637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.758270637\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.480120358979907257614801856925, −8.150023019607876992356235330815, −7.81689811978282778668797649272, −7.14359331369420660641505134271, −6.72859239757458973955457411080, −6.39534824627325212531410849263, −5.88574394927214672793628705247, −5.29419065603799323574299818472, −4.52665418236606087853371444935, −4.33807129117087452251507984995, −3.40442957639804775317817362043, −2.93217968193957477351947494572, −2.25707712581401728994510254043, −1.77289840913637707277420784339, −0.76288737653725960583109903584,
0.76288737653725960583109903584, 1.77289840913637707277420784339, 2.25707712581401728994510254043, 2.93217968193957477351947494572, 3.40442957639804775317817362043, 4.33807129117087452251507984995, 4.52665418236606087853371444935, 5.29419065603799323574299818472, 5.88574394927214672793628705247, 6.39534824627325212531410849263, 6.72859239757458973955457411080, 7.14359331369420660641505134271, 7.81689811978282778668797649272, 8.150023019607876992356235330815, 8.480120358979907257614801856925
