| L(s) = 1 | − 2·2-s + 3·4-s + 2·8-s − 9-s + 2·11-s − 3·16-s + 2·18-s − 4·22-s − 12·23-s − 14·25-s + 2·29-s + 12·32-s − 3·36-s − 20·37-s + 14·43-s + 6·44-s + 24·46-s + 28·50-s + 48·53-s − 4·58-s − 2·64-s − 4·67-s − 16·71-s − 2·72-s + 40·74-s − 24·79-s + 15·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.707·8-s − 1/3·9-s + 0.603·11-s − 3/4·16-s + 0.471·18-s − 0.852·22-s − 2.50·23-s − 2.79·25-s + 0.371·29-s + 2.12·32-s − 1/2·36-s − 3.28·37-s + 2.13·43-s + 0.904·44-s + 3.53·46-s + 3.95·50-s + 6.59·53-s − 0.525·58-s − 1/4·64-s − 0.488·67-s − 1.89·71-s − 0.235·72-s + 4.64·74-s − 2.70·79-s + 5/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.328902623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328902623\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + p T^{4} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 + T - 3 T^{3} - 5 T^{4} - 3 p T^{5} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 + T^{2} - 14 T^{4} - p T^{6} + 5 p^{3} T^{8} - p^{3} T^{10} - 14 p^{4} T^{12} + p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 + 7 T^{2} + 59 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - T + 8 T^{2} + 29 T^{3} - 83 T^{4} + 29 p T^{5} + 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 16 T^{2} + 251 T^{4} + 9168 T^{6} - 157480 T^{8} + 9168 p^{2} T^{10} + 251 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 63 T^{2} + 2258 T^{4} - 62307 T^{6} + 1364391 T^{8} - 62307 p^{2} T^{10} + 2258 p^{4} T^{12} - 63 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 6 T - 6 T^{2} - 24 T^{3} + 407 T^{4} - 24 p T^{5} - 6 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - T - 54 T^{2} + 3 T^{3} + 2155 T^{4} + 3 p T^{5} - 54 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 72 T^{2} + 2581 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 10 T + 14 T^{2} + 120 T^{3} + 2327 T^{4} + 120 p T^{5} + 14 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 8 T^{2} - 2846 T^{4} + 3616 T^{6} + 5534755 T^{8} + 3616 p^{2} T^{10} - 2846 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 7 T + 32 T^{2} + 483 T^{3} - 3667 T^{4} + 483 p T^{5} + 32 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 32 T^{2} - 1059 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 12 T + 129 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 - 184 T^{2} + 18755 T^{4} - 1497576 T^{6} + 98070104 T^{8} - 1497576 p^{2} T^{10} + 18755 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 192 T^{2} + 20258 T^{4} - 1759488 T^{6} + 124742451 T^{8} - 1759488 p^{2} T^{10} + 20258 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 240 T^{2} + 25006 T^{4} + 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 6 T + 154 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 280 T^{2} + 33053 T^{4} + 280 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 239 T^{2} + 27262 T^{4} - 3350063 T^{6} + 376362199 T^{8} - 3350063 p^{2} T^{10} + 27262 p^{4} T^{12} - 239 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 141 T^{2} + 74 T^{4} - 139449 T^{6} + 116727639 T^{8} - 139449 p^{2} T^{10} + 74 p^{4} T^{12} - 141 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.56728148578927855568062542163, −4.43859017682249900036757608598, −4.31366968624434947174578385894, −4.07281619270835394406863968169, −4.02865999138294273477824972901, −3.89277214277907347779548060582, −3.88842867203087388304476290008, −3.68960547837397041636009258577, −3.43497354338087349875385977757, −3.41924764994475015402629261393, −3.25949850134240769253431019742, −3.03620462983589682119790021823, −2.63077672105226735147720685904, −2.42441246894867687522291625691, −2.37829516638545847841447354908, −2.34319591011078898335992378749, −2.27767983567120718258852681783, −2.05412434316914290697350386237, −1.56284211725174995451881222743, −1.55673566926245904266820283336, −1.42916608395552063693088818392, −1.29433465208163791353622359503, −0.950068680988019740981576199563, −0.60362787116950152001889747997, −0.24180487227083459192103064091,
0.24180487227083459192103064091, 0.60362787116950152001889747997, 0.950068680988019740981576199563, 1.29433465208163791353622359503, 1.42916608395552063693088818392, 1.55673566926245904266820283336, 1.56284211725174995451881222743, 2.05412434316914290697350386237, 2.27767983567120718258852681783, 2.34319591011078898335992378749, 2.37829516638545847841447354908, 2.42441246894867687522291625691, 2.63077672105226735147720685904, 3.03620462983589682119790021823, 3.25949850134240769253431019742, 3.41924764994475015402629261393, 3.43497354338087349875385977757, 3.68960547837397041636009258577, 3.88842867203087388304476290008, 3.89277214277907347779548060582, 4.02865999138294273477824972901, 4.07281619270835394406863968169, 4.31366968624434947174578385894, 4.43859017682249900036757608598, 4.56728148578927855568062542163
Plot not available for L-functions of degree greater than 10.
