| L(s) = 1 | + 2-s − 2·3-s − 5-s − 2·6-s − 8-s + 3·9-s − 10-s − 10·11-s − 13-s + 2·15-s − 16-s + 4·17-s + 3·18-s + 5·19-s − 10·22-s + 2·23-s + 2·24-s − 26-s − 10·27-s − 4·29-s + 2·30-s − 8·31-s + 20·33-s + 4·34-s − 37-s + 5·38-s + 2·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 0.447·5-s − 0.816·6-s − 0.353·8-s + 9-s − 0.316·10-s − 3.01·11-s − 0.277·13-s + 0.516·15-s − 1/4·16-s + 0.970·17-s + 0.707·18-s + 1.14·19-s − 2.13·22-s + 0.417·23-s + 0.408·24-s − 0.196·26-s − 1.92·27-s − 0.742·29-s + 0.365·30-s − 1.43·31-s + 3.48·33-s + 0.685·34-s − 0.164·37-s + 0.811·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7509699745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7509699745\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 37 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T + 125 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 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
−11.76322199550366734772591323188, −11.06779008100379734760574420086, −10.84601578809610509013300348506, −10.25256200220773590864933332714, −10.25116668914895818659025633628, −9.215774089214437987349125913648, −9.163072464169665454155127985921, −7.974658163138372219247366183662, −7.82249238756354216837667163626, −7.22553558509920183455186415813, −7.19273424279266630867027149360, −5.85100978695002823088020972189, −5.71175849677635671421764861753, −5.21644606767308592998661612496, −5.11240325160584542569780058034, −4.18279680767359396339270186098, −3.60818994876474369599532308598, −2.88365581661015793084130787781, −2.13905486339130836933322406203, −0.54919039687456284305427886453,
0.54919039687456284305427886453, 2.13905486339130836933322406203, 2.88365581661015793084130787781, 3.60818994876474369599532308598, 4.18279680767359396339270186098, 5.11240325160584542569780058034, 5.21644606767308592998661612496, 5.71175849677635671421764861753, 5.85100978695002823088020972189, 7.19273424279266630867027149360, 7.22553558509920183455186415813, 7.82249238756354216837667163626, 7.974658163138372219247366183662, 9.163072464169665454155127985921, 9.215774089214437987349125913648, 10.25116668914895818659025633628, 10.25256200220773590864933332714, 10.84601578809610509013300348506, 11.06779008100379734760574420086, 11.76322199550366734772591323188
