| L(s) = 1 | + 4·3-s − 2·5-s + 10·9-s − 8·15-s + 10·17-s + 16·19-s − 25-s + 20·27-s + 12·31-s − 20·45-s + 21·49-s + 40·51-s + 64·57-s − 16·59-s + 10·61-s + 8·67-s − 16·71-s + 18·73-s − 4·75-s − 4·79-s + 35·81-s − 20·85-s + 48·93-s − 32·95-s − 36·101-s − 28·103-s − 8·107-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 10/3·9-s − 2.06·15-s + 2.42·17-s + 3.67·19-s − 1/5·25-s + 3.84·27-s + 2.15·31-s − 2.98·45-s + 3·49-s + 5.60·51-s + 8.47·57-s − 2.08·59-s + 1.28·61-s + 0.977·67-s − 1.89·71-s + 2.10·73-s − 0.461·75-s − 0.450·79-s + 35/9·81-s − 2.16·85-s + 4.97·93-s − 3.28·95-s − 3.58·101-s − 2.75·103-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.58148318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.58148318\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 200 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 576 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 350 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 66 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 426 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 6206 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 3648 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 169 T^{2} + 11484 T^{4} - 169 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 5 T + 120 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 - 9 T + 92 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.47816862044829957790709997018, −7.12123962796430343679173402027, −6.90134742053685752455567041882, −6.78216399252225813885329820980, −6.60983858287469009027381671001, −5.88534434443451320230515303027, −5.73396342925952404787992959077, −5.67341884168134899915176535878, −5.48589418835168856426897195962, −5.15326131536363447616618362452, −4.61260135922889494099416621981, −4.57177915664312458115144964108, −4.51037427257691017618844187201, −3.84426066843044352560593992723, −3.67597209665822873604286440429, −3.55220509058854867646776787379, −3.44257599549231737963584761857, −2.99547375848833738652827795031, −2.83491091067489463361957300006, −2.52634863201967623968541861589, −2.45199463372043304074658687339, −1.70162371498176958513907432921, −1.22838343902156966590316805602, −1.15485181654875614049428222535, −0.801958842338028592861741425657,
0.801958842338028592861741425657, 1.15485181654875614049428222535, 1.22838343902156966590316805602, 1.70162371498176958513907432921, 2.45199463372043304074658687339, 2.52634863201967623968541861589, 2.83491091067489463361957300006, 2.99547375848833738652827795031, 3.44257599549231737963584761857, 3.55220509058854867646776787379, 3.67597209665822873604286440429, 3.84426066843044352560593992723, 4.51037427257691017618844187201, 4.57177915664312458115144964108, 4.61260135922889494099416621981, 5.15326131536363447616618362452, 5.48589418835168856426897195962, 5.67341884168134899915176535878, 5.73396342925952404787992959077, 5.88534434443451320230515303027, 6.60983858287469009027381671001, 6.78216399252225813885329820980, 6.90134742053685752455567041882, 7.12123962796430343679173402027, 7.47816862044829957790709997018
