| L(s) = 1 | − 4·3-s − 8·7-s + 4·9-s − 12·19-s + 32·21-s + 12·25-s + 4·27-s − 16·37-s + 34·49-s + 48·57-s − 36·59-s − 32·63-s − 48·75-s − 10·81-s − 36·83-s − 48·103-s + 64·111-s − 24·113-s + 12·121-s + 127-s + 131-s + 96·133-s + 137-s + 139-s − 136·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3.02·7-s + 4/3·9-s − 2.75·19-s + 6.98·21-s + 12/5·25-s + 0.769·27-s − 2.63·37-s + 34/7·49-s + 6.35·57-s − 4.68·59-s − 4.03·63-s − 5.54·75-s − 1.11·81-s − 3.95·83-s − 4.72·103-s + 6.07·111-s − 2.25·113-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s − 11.2·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $D_{4}$ | \( ( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 426 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3414 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 18 T + 172 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 6186 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14646 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 11366 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 18 T + 244 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 7910 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.97600020581979586665913916341, −6.48725135068307125875538137542, −6.47871891668764475342300454620, −6.36690924885534247526547268336, −6.35966648052312360341328558092, −5.99900912365285347004724697219, −5.91697028629810257829481358012, −5.56871904035232570109482268732, −5.42232707337653219633245498629, −5.16363537316703180111373355706, −4.96325684790915315765906237501, −4.86331126143843380489271123861, −4.47421627353233459033464896071, −4.07081809185426965818807253880, −4.01087434653002081589728224025, −3.89073921561815405204825874530, −3.46114302633035693399811923989, −3.07123768203484336667637956924, −3.01797664367531999021518031612, −2.70153636148752191107252720063, −2.66044924043062550510326371169, −2.23793610966572628734204267791, −1.52572226427085762416301879395, −1.47946979426752302032837500276, −1.02635586464331361319313148619, 0, 0, 0, 0,
1.02635586464331361319313148619, 1.47946979426752302032837500276, 1.52572226427085762416301879395, 2.23793610966572628734204267791, 2.66044924043062550510326371169, 2.70153636148752191107252720063, 3.01797664367531999021518031612, 3.07123768203484336667637956924, 3.46114302633035693399811923989, 3.89073921561815405204825874530, 4.01087434653002081589728224025, 4.07081809185426965818807253880, 4.47421627353233459033464896071, 4.86331126143843380489271123861, 4.96325684790915315765906237501, 5.16363537316703180111373355706, 5.42232707337653219633245498629, 5.56871904035232570109482268732, 5.91697028629810257829481358012, 5.99900912365285347004724697219, 6.35966648052312360341328558092, 6.36690924885534247526547268336, 6.47871891668764475342300454620, 6.48725135068307125875538137542, 6.97600020581979586665913916341
