| L(s) = 1 | + 4·2-s + 4·3-s + 10·4-s + 16·6-s + 8·7-s + 20·8-s + 6·9-s + 40·12-s + 32·14-s + 35·16-s + 24·18-s + 32·21-s + 80·24-s − 2·25-s − 4·27-s + 80·28-s − 8·29-s + 56·32-s + 60·36-s − 16·41-s + 128·42-s + 32·43-s + 140·48-s + 16·49-s − 8·50-s − 24·53-s − 16·54-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s + 6.53·6-s + 3.02·7-s + 7.07·8-s + 2·9-s + 11.5·12-s + 8.55·14-s + 35/4·16-s + 5.65·18-s + 6.98·21-s + 16.3·24-s − 2/5·25-s − 0.769·27-s + 15.1·28-s − 1.48·29-s + 9.89·32-s + 10·36-s − 2.49·41-s + 19.7·42-s + 4.87·43-s + 20.2·48-s + 16/7·49-s − 1.13·50-s − 3.29·53-s − 2.17·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(60.28216530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(60.28216530\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 130 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 66 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 16 T + 148 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5766 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 11110 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 13942 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 34690 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80202242227694047962297532147, −7.60048286717238886568321299370, −7.41776250938688509094261352850, −7.05893811535262297355140259962, −6.93632721454390896411354400547, −6.15120139300849228076225475644, −6.13802179362875063779122290392, −6.07107195804993533841591465537, −5.82422646652259103790099859648, −5.33073275625658685155226248282, −5.13496602190129717167897367182, −4.83505278822222212966969056197, −4.72734722309184612335936219697, −4.48588244476838388878468021497, −4.19930328807414538843215952970, −4.06691870951529108305198035204, −3.51427677932967976688097641075, −3.29687294806212818937218886176, −3.19322445155388735948479048089, −2.94739214611423107813885716974, −2.39148028283665903808328174644, −2.02476914482624975678854849834, −1.98662185733131327668353988066, −1.62043900200395382227800313105, −1.39619237750898003875282140663,
1.39619237750898003875282140663, 1.62043900200395382227800313105, 1.98662185733131327668353988066, 2.02476914482624975678854849834, 2.39148028283665903808328174644, 2.94739214611423107813885716974, 3.19322445155388735948479048089, 3.29687294806212818937218886176, 3.51427677932967976688097641075, 4.06691870951529108305198035204, 4.19930328807414538843215952970, 4.48588244476838388878468021497, 4.72734722309184612335936219697, 4.83505278822222212966969056197, 5.13496602190129717167897367182, 5.33073275625658685155226248282, 5.82422646652259103790099859648, 6.07107195804993533841591465537, 6.13802179362875063779122290392, 6.15120139300849228076225475644, 6.93632721454390896411354400547, 7.05893811535262297355140259962, 7.41776250938688509094261352850, 7.60048286717238886568321299370, 7.80202242227694047962297532147
