| L(s) = 1 | + 16·7-s + 4·9-s + 184·17-s − 16·23-s + 108·25-s − 768·31-s + 600·41-s − 32·47-s − 828·49-s + 64·63-s − 816·71-s − 824·73-s − 800·79-s − 678·81-s − 1.14e3·89-s + 4.40e3·97-s + 6.92e3·103-s + 392·113-s + 2.94e3·119-s + 996·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 736·153-s + ⋯ |
| L(s) = 1 | + 0.863·7-s + 4/27·9-s + 2.62·17-s − 0.145·23-s + 0.863·25-s − 4.44·31-s + 2.28·41-s − 0.0993·47-s − 2.41·49-s + 0.127·63-s − 1.36·71-s − 1.32·73-s − 1.13·79-s − 0.930·81-s − 1.36·89-s + 4.61·97-s + 6.62·103-s + 0.326·113-s + 2.26·119-s + 0.748·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.388·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.846499832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.846499832\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 694 T^{4} - 4 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 31094 T^{4} - 108 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 8 T + 510 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 996 T^{2} + 27974 p^{2} T^{4} - 996 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 332 T^{2} - 7598826 T^{4} - 332 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 92 T + 5030 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1196 p T^{2} + 223149174 T^{4} - 1196 p^{7} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 8 T + 8798 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 84428 T^{2} + 2937799638 T^{4} - 84428 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 384 T + 84158 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 169004 T^{2} + 11993714550 T^{4} - 169004 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 300 T + 157270 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 270628 T^{2} + 30844604694 T^{4} - 270628 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 16 T + 114782 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 576620 T^{2} + 127450023606 T^{4} - 576620 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 509604 T^{2} + 127606046294 T^{4} - 509604 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 394892 T^{2} + 121971578838 T^{4} - 394892 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 526916 T^{2} + 240544281654 T^{4} - 526916 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 408 T + 672766 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 412 T + 690678 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 400 T + 963870 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1821572 T^{2} + 1476121597686 T^{4} - 1821572 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 572 T + 845846 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 2204 T + 2633478 T^{2} - 2204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−8.144989756438583839044631393367, −7.909620961497077057998670253811, −7.74273651253896813153679048917, −7.59419399778250870121563942415, −7.22063961086678029016042958605, −7.10542638423476562296446841520, −7.00325267386804052595945813729, −6.10788403549982966185195191672, −5.99866826119698498683691037349, −5.95206834216582631959586549465, −5.70413441068082803925731515823, −5.19642169023284438437979972884, −4.96098072687574196728508073086, −4.72739685823402386405169901815, −4.54508503783179506804028198496, −3.76729511820641267251786936970, −3.71002657950735180142754015905, −3.49781267979380383192460404522, −2.97898206473840872037561366224, −2.76934359926576829859527878266, −2.00513844915619788274730023244, −1.66753405737600737301223170695, −1.53656274133624747750135255868, −0.868209664837127757118532512506, −0.38486846916063418233492099180,
0.38486846916063418233492099180, 0.868209664837127757118532512506, 1.53656274133624747750135255868, 1.66753405737600737301223170695, 2.00513844915619788274730023244, 2.76934359926576829859527878266, 2.97898206473840872037561366224, 3.49781267979380383192460404522, 3.71002657950735180142754015905, 3.76729511820641267251786936970, 4.54508503783179506804028198496, 4.72739685823402386405169901815, 4.96098072687574196728508073086, 5.19642169023284438437979972884, 5.70413441068082803925731515823, 5.95206834216582631959586549465, 5.99866826119698498683691037349, 6.10788403549982966185195191672, 7.00325267386804052595945813729, 7.10542638423476562296446841520, 7.22063961086678029016042958605, 7.59419399778250870121563942415, 7.74273651253896813153679048917, 7.909620961497077057998670253811, 8.144989756438583839044631393367
