| L(s) = 1 | − 16-s − 4·25-s − 2·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 16-s − 4·25-s − 2·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2893713329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2893713329\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−9.359459523501214936299721487488, −8.616183523062858570048153568749, −8.430249922488496340160965451890, −8.359760427737248794567555870120, −8.052452622318778430516936318975, −7.60362404025781515972335137591, −7.53100868017079516750796325110, −7.37636580278176534769287781777, −6.97169499829725454542960321948, −6.38076241734211654654451308769, −6.38010261039838889832853376282, −6.34382723285042271943705451271, −5.70558301503066036626691818457, −5.59473767159691072289295351651, −5.20646784822708528484804550679, −5.01365944078172667346786324968, −4.37519698725907040728084771910, −4.19563333390428612570750601132, −4.15660480590871533776607540093, −3.48611782702902368647548111112, −3.33686108484465001866390731047, −2.83245713735765813822820510684, −2.20863996460587219645669854425, −2.00353902930501664458134600861, −1.56459043903912568487343958848,
1.56459043903912568487343958848, 2.00353902930501664458134600861, 2.20863996460587219645669854425, 2.83245713735765813822820510684, 3.33686108484465001866390731047, 3.48611782702902368647548111112, 4.15660480590871533776607540093, 4.19563333390428612570750601132, 4.37519698725907040728084771910, 5.01365944078172667346786324968, 5.20646784822708528484804550679, 5.59473767159691072289295351651, 5.70558301503066036626691818457, 6.34382723285042271943705451271, 6.38010261039838889832853376282, 6.38076241734211654654451308769, 6.97169499829725454542960321948, 7.37636580278176534769287781777, 7.53100868017079516750796325110, 7.60362404025781515972335137591, 8.052452622318778430516936318975, 8.359760427737248794567555870120, 8.430249922488496340160965451890, 8.616183523062858570048153568749, 9.359459523501214936299721487488
