| L(s) = 1 | + 4·2-s − 2·3-s + 10·4-s − 8·6-s + 20·8-s + 5·9-s − 20·12-s + 35·16-s − 6·17-s + 20·18-s − 40·24-s − 16·27-s + 8·29-s + 56·32-s − 24·34-s + 50·36-s + 32·37-s + 10·43-s − 20·47-s − 70·48-s + 49-s + 12·51-s − 64·54-s + 32·58-s + 46·59-s + 84·64-s − 60·68-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 1.15·3-s + 5·4-s − 3.26·6-s + 7.07·8-s + 5/3·9-s − 5.77·12-s + 35/4·16-s − 1.45·17-s + 4.71·18-s − 8.16·24-s − 3.07·27-s + 1.48·29-s + 9.89·32-s − 4.11·34-s + 25/3·36-s + 5.26·37-s + 1.52·43-s − 2.91·47-s − 10.1·48-s + 1/7·49-s + 1.68·51-s − 8.70·54-s + 4.20·58-s + 5.98·59-s + 21/2·64-s − 7.27·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(23.98728632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.98728632\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 3 | $D_{4}$ | \( ( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_4\times C_2$ | \( 1 - T^{2} - 83 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 33 T^{2} + 545 T^{4} - 33 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 2533 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 97 T^{2} + 4093 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 4254 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 5 T + 85 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 137 T^{2} + 9433 T^{4} - 137 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 23 T + 243 T^{2} - 23 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 9 T^{2} + 4265 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6430 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 93 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 51 T^{2} + 11501 T^{4} + 51 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 14038 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 296 T^{2} + 37630 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 7 T + 141 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.53112246086833917106840051637, −6.37978820389973195022399852235, −6.35031553327905355404368479591, −6.06092736367907227073145029596, −6.01334823876575463578065429801, −5.56387826433835846685348055957, −5.42158544710518720275502868449, −5.34513792999938993423385914558, −4.91315372926070235542198289524, −4.73812945187141107382561046748, −4.53520533664483401948006802959, −4.48519536874195823232200957709, −4.12967060007767316994399878911, −4.01808116533259794248874455033, −3.74520821320496714596400099984, −3.55550535201040755309634703016, −3.27330892237816560760503112530, −2.71019647421272720265357069746, −2.48627634940957361304060630276, −2.48197379254146863351010534813, −2.08485618461288717128328743404, −1.92813292952257738444075624059, −1.15927659919227557852237169594, −1.04196405087624144964722072195, −0.61531777419852440943898378862,
0.61531777419852440943898378862, 1.04196405087624144964722072195, 1.15927659919227557852237169594, 1.92813292952257738444075624059, 2.08485618461288717128328743404, 2.48197379254146863351010534813, 2.48627634940957361304060630276, 2.71019647421272720265357069746, 3.27330892237816560760503112530, 3.55550535201040755309634703016, 3.74520821320496714596400099984, 4.01808116533259794248874455033, 4.12967060007767316994399878911, 4.48519536874195823232200957709, 4.53520533664483401948006802959, 4.73812945187141107382561046748, 4.91315372926070235542198289524, 5.34513792999938993423385914558, 5.42158544710518720275502868449, 5.56387826433835846685348055957, 6.01334823876575463578065429801, 6.06092736367907227073145029596, 6.35031553327905355404368479591, 6.37978820389973195022399852235, 6.53112246086833917106840051637
