| L(s) = 1 | − 2·2-s + 5·5-s + 4·7-s + 8·8-s − 10·10-s + 48·11-s − 2·13-s − 8·14-s − 16·16-s − 228·17-s + 280·19-s − 96·22-s − 72·23-s + 4·26-s − 210·29-s − 272·31-s + 456·34-s + 20·35-s − 668·37-s − 560·38-s + 40·40-s + 198·41-s + 268·43-s + 144·46-s − 216·47-s + 343·49-s − 156·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.447·5-s + 0.215·7-s + 0.353·8-s − 0.316·10-s + 1.31·11-s − 0.0426·13-s − 0.152·14-s − 1/4·16-s − 3.25·17-s + 3.38·19-s − 0.930·22-s − 0.652·23-s + 0.0301·26-s − 1.34·29-s − 1.57·31-s + 2.30·34-s + 0.0965·35-s − 2.96·37-s − 2.39·38-s + 0.158·40-s + 0.754·41-s + 0.950·43-s + 0.461·46-s − 0.670·47-s + 49-s − 0.404·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4868536288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4868536288\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T - 327 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 48 T + 973 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T - 2193 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 114 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 140 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 72 T - 6983 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 210 T + 19711 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 272 T + 44193 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 334 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 198 T - 29717 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 268 T - 7683 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 216 T - 57167 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 78 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 240 T - 147779 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 302 T - 135777 T^{2} + 302 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 596 T + 54453 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 768 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 478 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 640 T - 83439 T^{2} - 640 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 348 T - 450683 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 210 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1534 T + 1440483 T^{2} - 1534 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.948803142664595719246401922463, −9.294011631793388166509074229030, −9.250988954048808336708468966002, −8.984159280521740295314032792267, −8.737245772324103545045651916639, −7.81110752509252653158896263146, −7.53731043249234003750357538343, −7.04557397410445080895579418777, −6.85223204447930698287103087803, −6.20675495835752627143755411642, −5.52004068151849288517290720147, −5.42876452954204143358306610172, −4.59802095882482836037123259004, −4.21461360619803631157018387916, −3.63328139636967747284214443138, −3.09250217730309355128884354598, −2.23787650472173326400753020356, −1.59847307293441694674770582672, −1.39266072303511398267761898096, −0.20970422872946751674271682152,
0.20970422872946751674271682152, 1.39266072303511398267761898096, 1.59847307293441694674770582672, 2.23787650472173326400753020356, 3.09250217730309355128884354598, 3.63328139636967747284214443138, 4.21461360619803631157018387916, 4.59802095882482836037123259004, 5.42876452954204143358306610172, 5.52004068151849288517290720147, 6.20675495835752627143755411642, 6.85223204447930698287103087803, 7.04557397410445080895579418777, 7.53731043249234003750357538343, 7.81110752509252653158896263146, 8.737245772324103545045651916639, 8.984159280521740295314032792267, 9.250988954048808336708468966002, 9.294011631793388166509074229030, 9.948803142664595719246401922463
