| L(s) = 1 | − 5·9-s − 14·11-s + 4·19-s + 6·29-s + 32·31-s + 4·41-s − 2·49-s + 32·59-s + 12·61-s − 32·71-s + 26·79-s + 9·81-s − 36·89-s + 70·99-s − 24·101-s + 10·109-s + 95·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 31·169-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 4.22·11-s + 0.917·19-s + 1.11·29-s + 5.74·31-s + 0.624·41-s − 2/7·49-s + 4.16·59-s + 1.53·61-s − 3.79·71-s + 2.92·79-s + 81-s − 3.81·89-s + 7.03·99-s − 2.38·101-s + 0.957·109-s + 8.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.38·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.950916178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.950916178\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 504 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 39 T^{2} + 752 T^{4} - 39 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 1070 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 4446 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 35 T^{2} + 4056 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4622 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 13 T + 126 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 12566 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 65 T^{2} + 3168 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.20501520538392712539823137475, −6.07904629571994722238486946142, −5.72779149384743419273173714540, −5.42997826449242752671567670548, −5.42136441412503553949907368158, −5.33341908392697770786119814894, −5.00060273119352482077339867766, −4.87017041580825592985359575967, −4.86356417594650151874217664225, −4.25378979804404486052084768788, −4.20164513627272831405351659815, −4.13239459449948265978111913769, −3.69845581074718933780520482144, −3.20554071939951152906990224633, −2.95625592431760719473249157258, −2.80232905571079184764088319446, −2.76675840188262147667750655324, −2.74391258124255383160731407580, −2.48788500986812673120362957120, −2.15969859668736097405843047206, −1.83827938349319940443775460060, −1.05782134401526671583367312490, −1.05176042708463144585098909372, −0.55373281022801603133583532460, −0.33508315801038059692740921018,
0.33508315801038059692740921018, 0.55373281022801603133583532460, 1.05176042708463144585098909372, 1.05782134401526671583367312490, 1.83827938349319940443775460060, 2.15969859668736097405843047206, 2.48788500986812673120362957120, 2.74391258124255383160731407580, 2.76675840188262147667750655324, 2.80232905571079184764088319446, 2.95625592431760719473249157258, 3.20554071939951152906990224633, 3.69845581074718933780520482144, 4.13239459449948265978111913769, 4.20164513627272831405351659815, 4.25378979804404486052084768788, 4.86356417594650151874217664225, 4.87017041580825592985359575967, 5.00060273119352482077339867766, 5.33341908392697770786119814894, 5.42136441412503553949907368158, 5.42997826449242752671567670548, 5.72779149384743419273173714540, 6.07904629571994722238486946142, 6.20501520538392712539823137475
