| L(s) = 1 | + 13·4-s − 36·7-s + 57·16-s − 84·19-s + 80·25-s − 468·28-s + 604·31-s − 184·37-s + 880·43-s − 96·49-s + 656·61-s + 117·64-s − 3.05e3·67-s + 312·73-s − 1.09e3·76-s − 720·79-s + 344·97-s + 1.04e3·100-s + 3.39e3·103-s − 3.38e3·109-s − 2.05e3·112-s − 4.57e3·121-s + 7.85e3·124-s + 127-s + 131-s + 3.02e3·133-s + 137-s + ⋯ |
| L(s) = 1 | + 13/8·4-s − 1.94·7-s + 0.890·16-s − 1.01·19-s + 0.639·25-s − 3.15·28-s + 3.49·31-s − 0.817·37-s + 3.12·43-s − 0.279·49-s + 1.37·61-s + 0.228·64-s − 5.56·67-s + 0.500·73-s − 1.64·76-s − 1.02·79-s + 0.360·97-s + 1.03·100-s + 3.24·103-s − 2.97·109-s − 1.73·112-s − 3.43·121-s + 5.68·124-s + 0.000698·127-s + 0.000666·131-s + 1.97·133-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6398019709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6398019709\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 - 13 T^{2} + 7 p^{4} T^{4} - 13 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 16 p T^{2} + 24462 T^{4} - 16 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 18 T + 534 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 4572 T^{2} + 8634710 T^{4} + 4572 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 16772 T^{2} + 118361542 T^{4} + 16772 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 42 T + 2742 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 12 T^{2} - 393658 T^{4} + 12 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 22356 T^{2} - 27485674 T^{4} + 22356 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 302 T + 2650 p T^{2} - 302 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 92 T + 43774 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 227840 T^{2} + 22474730590 T^{4} + 227840 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 440 T + 203686 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 270652 T^{2} + 39420081222 T^{4} + 270652 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 438372 T^{2} + 89997219542 T^{4} + 438372 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 598556 T^{2} + 161876495254 T^{4} + 598556 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 328 T + 368086 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 1526 T + 1116358 T^{2} + 1526 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 500316 T^{2} + 133803927398 T^{4} + 500316 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 156 T - 293274 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 360 T + 287790 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 2272508 T^{2} + 1944939568822 T^{4} + 2272508 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 913024 T^{2} + 920290760478 T^{4} + 913024 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 172 T + 1202710 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.35757822308983854382980798207, −6.25851941313711178118809708893, −6.11741343174577976432542809306, −5.87417981942473984814014811701, −5.61496657885029602100652748297, −5.48600051285457318152492303167, −4.87454759940361336813191133018, −4.86578790436755619937895737924, −4.57908819378458664304511674055, −4.31575649839361882174505172912, −4.19349295601909716129310325048, −3.85756569764793113997889899285, −3.48863132100031582966975976634, −3.34331915764237245444414103240, −2.98277169843247097047485881714, −2.81550082141436203728273721138, −2.67741733400289079674842183723, −2.49331004193459857867638328145, −2.26138110203357270172048365855, −1.74259041634963410151916401331, −1.58434371322442796836646098508, −1.14788577304133737792447267803, −0.911963064890950352177229371527, −0.53022959725698598052484078048, −0.087232641167494068749676782804,
0.087232641167494068749676782804, 0.53022959725698598052484078048, 0.911963064890950352177229371527, 1.14788577304133737792447267803, 1.58434371322442796836646098508, 1.74259041634963410151916401331, 2.26138110203357270172048365855, 2.49331004193459857867638328145, 2.67741733400289079674842183723, 2.81550082141436203728273721138, 2.98277169843247097047485881714, 3.34331915764237245444414103240, 3.48863132100031582966975976634, 3.85756569764793113997889899285, 4.19349295601909716129310325048, 4.31575649839361882174505172912, 4.57908819378458664304511674055, 4.86578790436755619937895737924, 4.87454759940361336813191133018, 5.48600051285457318152492303167, 5.61496657885029602100652748297, 5.87417981942473984814014811701, 6.11741343174577976432542809306, 6.25851941313711178118809708893, 6.35757822308983854382980798207
