| 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
| \(L(1)\) |
\(\approx\) |
\(0.5969304808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5969304808\) |
| \(L(\frac{3}{2})\) |
|
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
−7.14839097121117454128937685430, −7.14573871654818016920316633707, −6.63740818499600890420239376467, −6.54178939955786328335063981814, −6.37993078613689624851210645357, −6.02733441165597819252156232452, −5.79677153032710855568665066029, −5.63505543411522152711353015088, −5.47645914520473065222480776230, −5.17404992056839283635588640041, −5.10485937446315768105789629432, −4.68607500326294385003141478683, −4.33359954585422204844628530313, −4.21989022528854330677031751615, −3.87052911292979782953202490879, −3.81026836997235737790763411114, −3.55568424868120638339187717100, −3.30120663720140239022297223840, −2.55023698786691310356092210960, −2.24759979608745054011809640632, −2.13583013763722260368947043397, −1.67793217684870770229711746270, −1.11186385128181932382564624403, −0.69400133832863919491635967668, −0.35143735854871599649762726020,
0.35143735854871599649762726020, 0.69400133832863919491635967668, 1.11186385128181932382564624403, 1.67793217684870770229711746270, 2.13583013763722260368947043397, 2.24759979608745054011809640632, 2.55023698786691310356092210960, 3.30120663720140239022297223840, 3.55568424868120638339187717100, 3.81026836997235737790763411114, 3.87052911292979782953202490879, 4.21989022528854330677031751615, 4.33359954585422204844628530313, 4.68607500326294385003141478683, 5.10485937446315768105789629432, 5.17404992056839283635588640041, 5.47645914520473065222480776230, 5.63505543411522152711353015088, 5.79677153032710855568665066029, 6.02733441165597819252156232452, 6.37993078613689624851210645357, 6.54178939955786328335063981814, 6.63740818499600890420239376467, 7.14573871654818016920316633707, 7.14839097121117454128937685430
