| L(s) = 1 | − 4·7-s − 12·11-s − 8·13-s + 7·16-s − 8·19-s + 4·31-s − 8·37-s + 12·41-s − 24·43-s − 12·47-s + 8·49-s − 12·53-s − 12·59-s + 12·61-s + 4·67-s − 24·71-s + 16·73-s + 48·77-s − 24·79-s − 9·81-s − 12·83-s − 12·89-s + 32·91-s − 16·97-s + 12·101-s + 48·103-s + 24·107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 3.61·11-s − 2.21·13-s + 7/4·16-s − 1.83·19-s + 0.718·31-s − 1.31·37-s + 1.87·41-s − 3.65·43-s − 1.75·47-s + 8/7·49-s − 1.64·53-s − 1.56·59-s + 1.53·61-s + 0.488·67-s − 2.84·71-s + 1.87·73-s + 5.47·77-s − 2.70·79-s − 81-s − 1.31·83-s − 1.27·89-s + 3.35·91-s − 1.62·97-s + 1.19·101-s + 4.72·103-s + 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2677283988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2677283988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - 7 T^{4} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 24 T^{3} + 71 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 312 T^{3} + 1127 T^{4} + 312 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 434 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1443 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 168 T^{3} - 1801 T^{4} + 168 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 264 T^{3} + 2162 T^{4} + 264 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 564 T^{3} + 4382 T^{4} - 564 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 744 T^{3} + 7463 T^{4} + 744 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 600 T^{3} + 4919 T^{4} + 600 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 168 T^{3} + 2903 T^{4} - 168 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1488 T^{3} + 16898 T^{4} - 1488 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 188 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 1176 T^{3} + 18983 T^{4} + 1176 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 1140 T^{3} + 18014 T^{4} + 1140 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1872 T^{3} + 26978 T^{4} + 1872 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.16418532200008140029059080580, −7.16114905587822706526718390762, −6.66917141310837032889927268426, −6.56906257707455093133996842248, −6.10348390774193006426390079071, −6.09226243802552784723112865693, −5.80250087978027956742838597312, −5.55907773864268851703545125827, −5.42207548031265714044616966611, −5.01290332540202589919105359734, −4.80899033076057623214112545874, −4.69929006384598797659268060175, −4.55727122312829571040021809585, −4.24842351555835634186738257387, −3.68434780819999038816414298316, −3.22911595922423436562872280906, −3.15019021800781514212774328229, −3.06514957820134689629114614229, −2.96897224568912511071206284077, −2.26820093267117406573087566811, −2.25700657961325171417968265731, −1.91757847175043990134455885967, −1.48517018543987007852812245286, −0.43982905600099564269196765310, −0.23180415170537481995940450780,
0.23180415170537481995940450780, 0.43982905600099564269196765310, 1.48517018543987007852812245286, 1.91757847175043990134455885967, 2.25700657961325171417968265731, 2.26820093267117406573087566811, 2.96897224568912511071206284077, 3.06514957820134689629114614229, 3.15019021800781514212774328229, 3.22911595922423436562872280906, 3.68434780819999038816414298316, 4.24842351555835634186738257387, 4.55727122312829571040021809585, 4.69929006384598797659268060175, 4.80899033076057623214112545874, 5.01290332540202589919105359734, 5.42207548031265714044616966611, 5.55907773864268851703545125827, 5.80250087978027956742838597312, 6.09226243802552784723112865693, 6.10348390774193006426390079071, 6.56906257707455093133996842248, 6.66917141310837032889927268426, 7.16114905587822706526718390762, 7.16418532200008140029059080580
