| L(s) = 1 | − 2·2-s − 11·4-s + 20·5-s + 36·8-s − 40·10-s + 20·11-s − 104·13-s + 61·16-s + 116·17-s − 192·19-s − 220·20-s − 40·22-s − 28·23-s + 148·25-s + 208·26-s − 296·29-s + 104·31-s − 358·32-s − 232·34-s − 248·37-s + 384·38-s + 720·40-s + 20·41-s − 720·43-s − 220·44-s + 56·46-s − 96·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.37·4-s + 1.78·5-s + 1.59·8-s − 1.26·10-s + 0.548·11-s − 2.21·13-s + 0.953·16-s + 1.65·17-s − 2.31·19-s − 2.45·20-s − 0.387·22-s − 0.253·23-s + 1.18·25-s + 1.56·26-s − 1.89·29-s + 0.602·31-s − 1.97·32-s − 1.17·34-s − 1.10·37-s + 1.63·38-s + 2.84·40-s + 0.0761·41-s − 2.55·43-s − 0.753·44-s + 0.179·46-s − 0.297·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 15 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 4 p T + 252 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 20 T + 1610 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 p T + 5848 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 116 T + 9140 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 192 T + 21966 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 28 T + 22962 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 296 T + 62994 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 104 T + 57286 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 248 T + 112074 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 20 T + 42020 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 720 T + 268614 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 96 T + 84950 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 268 T + 56510 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 616 T + 403470 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 454008 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 144 T + 57558 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 988 T + 852210 T^{2} + 988 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 104 T + 459136 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 944 T + 1097470 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1016 T + 1388838 T^{2} - 1016 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 388 T + 1217732 T^{2} + 388 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 488 T + 1167280 T^{2} + 488 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.25115140680928069793907822682, −9.969466995847684466964555341501, −9.476274308523786169016516636709, −9.424361388081104139528733568921, −8.724496899951372790870647962732, −8.474033469277892391220969697937, −7.63773511248203626881775284757, −7.59559597392885505268808199278, −6.62161073278584738876393138224, −6.31024770635862765524137790211, −5.42997434334579111917571478556, −5.36998073281495201553786291057, −4.71367889827278896047239244691, −4.23526549429974276304580344014, −3.48169475595731462859201874757, −2.62248657262314888058741630356, −1.67612468822529134862330873373, −1.61028121729515127856046761135, 0, 0,
1.61028121729515127856046761135, 1.67612468822529134862330873373, 2.62248657262314888058741630356, 3.48169475595731462859201874757, 4.23526549429974276304580344014, 4.71367889827278896047239244691, 5.36998073281495201553786291057, 5.42997434334579111917571478556, 6.31024770635862765524137790211, 6.62161073278584738876393138224, 7.59559597392885505268808199278, 7.63773511248203626881775284757, 8.474033469277892391220969697937, 8.724496899951372790870647962732, 9.424361388081104139528733568921, 9.476274308523786169016516636709, 9.969466995847684466964555341501, 10.25115140680928069793907822682
