| L(s) = 1 | − 4·3-s − 3·4-s − 2·9-s + 12·12-s + 4·16-s + 40·27-s + 6·36-s + 24·43-s − 16·48-s − 22·49-s − 12·53-s − 24·61-s − 9·64-s − 24·79-s − 55·81-s − 36·101-s − 24·103-s + 24·107-s − 120·108-s + 48·113-s − 30·121-s + 127-s − 96·129-s + 131-s + 137-s + 139-s − 8·144-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3/2·4-s − 2/3·9-s + 3.46·12-s + 16-s + 7.69·27-s + 36-s + 3.65·43-s − 2.30·48-s − 3.14·49-s − 1.64·53-s − 3.07·61-s − 9/8·64-s − 2.70·79-s − 6.11·81-s − 3.58·101-s − 2.36·103-s + 2.32·107-s − 11.5·108-s + 4.51·113-s − 2.72·121-s + 0.0887·127-s − 8.45·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2/3·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 5 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 1182 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 75 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 108 T^{2} + 5990 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 30 T^{2} - 1213 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 46 T^{2} - 2589 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 222 T^{2} + 22067 T^{4} + 222 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 + 168 T^{2} + 18734 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 246 T^{2} + 29627 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 238 T^{2} + 29955 T^{4} + 238 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.16072673295501912114632913642, −5.98500167427428650822005883122, −5.90464586975854880484947871814, −5.74127861722261324399473562782, −5.69204871831171497937169622451, −5.23287928108012956365690741149, −5.22171129909107855231853300001, −4.99113476206321493939510200968, −4.86863507100071047886778358060, −4.58837090644337416749002677574, −4.41692300216820867732553391384, −4.39064573423393449984918980883, −4.17623590961592529021757109377, −3.60526153508784999008247464024, −3.53934252396509762949591527742, −3.29244026978057639255724617659, −3.01891831970089841193363721930, −2.90547740533077800396688692009, −2.58082788530152075936552590618, −2.53546772063865558895386522329, −2.11763638514536499094369540955, −1.66858814157829210599600190681, −1.21538458339839894044951839893, −1.05851626593819074887672391877, −0.949847156161175933242480810665, 0, 0, 0, 0,
0.949847156161175933242480810665, 1.05851626593819074887672391877, 1.21538458339839894044951839893, 1.66858814157829210599600190681, 2.11763638514536499094369540955, 2.53546772063865558895386522329, 2.58082788530152075936552590618, 2.90547740533077800396688692009, 3.01891831970089841193363721930, 3.29244026978057639255724617659, 3.53934252396509762949591527742, 3.60526153508784999008247464024, 4.17623590961592529021757109377, 4.39064573423393449984918980883, 4.41692300216820867732553391384, 4.58837090644337416749002677574, 4.86863507100071047886778358060, 4.99113476206321493939510200968, 5.22171129909107855231853300001, 5.23287928108012956365690741149, 5.69204871831171497937169622451, 5.74127861722261324399473562782, 5.90464586975854880484947871814, 5.98500167427428650822005883122, 6.16072673295501912114632913642
