| L(s) = 1 | + 2-s − 3-s + 3·5-s − 6-s − 8-s + 3·9-s + 3·10-s − 6·11-s + 2·13-s − 3·15-s − 16-s + 3·17-s + 3·18-s − 2·19-s − 6·22-s + 24-s + 5·25-s + 2·26-s − 8·27-s + 12·29-s − 3·30-s + 4·31-s + 6·33-s + 3·34-s + 7·37-s − 2·38-s − 2·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1.34·5-s − 0.408·6-s − 0.353·8-s + 9-s + 0.948·10-s − 1.80·11-s + 0.554·13-s − 0.774·15-s − 1/4·16-s + 0.727·17-s + 0.707·18-s − 0.458·19-s − 1.27·22-s + 0.204·24-s + 25-s + 0.392·26-s − 1.53·27-s + 2.22·29-s − 0.547·30-s + 0.718·31-s + 1.04·33-s + 0.514·34-s + 1.15·37-s − 0.324·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1623076 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1623076 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.075908850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.075908850\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.02414614354373510125618424346, −9.642144663484840962972623906659, −9.214793479021470869671545744802, −8.601675652314400376932627301865, −8.245821431173540695722316945748, −7.80803129834359095405016379425, −7.40211630618632686499206078777, −6.83349163197426809178972307291, −6.34418893371332079710871552929, −5.94716665172481786328718230579, −5.85522540598228304962787716990, −5.10217242807947181692504087355, −4.97070784315375346081156658939, −4.44416306589418593545210854697, −4.00602453723288286774536932848, −3.00555991530369871067846111712, −2.93100660601375220566987741143, −2.15382621102008754643720611810, −1.53712857088725695217922873413, −0.69911432621316140419650129518,
0.69911432621316140419650129518, 1.53712857088725695217922873413, 2.15382621102008754643720611810, 2.93100660601375220566987741143, 3.00555991530369871067846111712, 4.00602453723288286774536932848, 4.44416306589418593545210854697, 4.97070784315375346081156658939, 5.10217242807947181692504087355, 5.85522540598228304962787716990, 5.94716665172481786328718230579, 6.34418893371332079710871552929, 6.83349163197426809178972307291, 7.40211630618632686499206078777, 7.80803129834359095405016379425, 8.245821431173540695722316945748, 8.601675652314400376932627301865, 9.214793479021470869671545744802, 9.642144663484840962972623906659, 10.02414614354373510125618424346
