| L(s) = 1 | − 6·3-s − 20·5-s + 27·9-s + 20·11-s − 104·13-s + 120·15-s − 116·17-s + 192·19-s − 28·23-s + 148·25-s − 108·27-s + 296·29-s − 104·31-s − 120·33-s − 248·37-s + 624·39-s − 20·41-s + 720·43-s − 540·45-s − 96·47-s + 696·51-s + 268·53-s − 400·55-s − 1.15e3·57-s − 616·59-s − 16·61-s + 2.08e3·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 9-s + 0.548·11-s − 2.21·13-s + 2.06·15-s − 1.65·17-s + 2.31·19-s − 0.253·23-s + 1.18·25-s − 0.769·27-s + 1.89·29-s − 0.602·31-s − 0.633·33-s − 1.10·37-s + 2.56·39-s − 0.0761·41-s + 2.55·43-s − 1.78·45-s − 0.297·47-s + 1.91·51-s + 0.694·53-s − 0.980·55-s − 2.67·57-s − 1.35·59-s − 0.0335·61-s + 3.96·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 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
−8.313848049333632306757232578255, −7.898997975686862013606797065162, −7.49687571464462253156407366739, −7.27021255194847001970482923685, −7.05283319900711838563717376269, −6.68594000746668801681336151486, −6.00381531244504878724641929374, −5.77658190782031829464222694122, −5.02228339916917045622774538424, −4.88246375787487243494738098897, −4.40423799531124633245909996196, −4.31672928450321038007033513879, −3.42279087325664384098749913940, −3.36403122522750713576301559803, −2.47201365552535200657278568105, −2.17500084842736339701338460257, −1.16128508964478700119890521665, −0.790406109945933608839055154648, 0, 0,
0.790406109945933608839055154648, 1.16128508964478700119890521665, 2.17500084842736339701338460257, 2.47201365552535200657278568105, 3.36403122522750713576301559803, 3.42279087325664384098749913940, 4.31672928450321038007033513879, 4.40423799531124633245909996196, 4.88246375787487243494738098897, 5.02228339916917045622774538424, 5.77658190782031829464222694122, 6.00381531244504878724641929374, 6.68594000746668801681336151486, 7.05283319900711838563717376269, 7.27021255194847001970482923685, 7.49687571464462253156407366739, 7.898997975686862013606797065162, 8.313848049333632306757232578255
