| L(s) = 1 | − 2·4-s − 2·9-s + 8·11-s − 5·16-s − 8·19-s + 8·29-s + 24·31-s + 4·36-s − 8·41-s − 16·44-s − 2·49-s − 8·61-s + 20·64-s + 40·71-s + 16·76-s − 16·79-s + 3·81-s + 8·89-s − 16·99-s − 56·101-s + 8·109-s − 16·116-s + 36·121-s − 48·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 2.41·11-s − 5/4·16-s − 1.83·19-s + 1.48·29-s + 4.31·31-s + 2/3·36-s − 1.24·41-s − 2.41·44-s − 2/7·49-s − 1.02·61-s + 5/2·64-s + 4.74·71-s + 1.83·76-s − 1.80·79-s + 1/3·81-s + 0.847·89-s − 1.60·99-s − 5.57·101-s + 0.766·109-s − 1.48·116-s + 3.27·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.640313712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.640313712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 1558 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 698 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 982 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 9382 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 6218 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 316 T^{2} + 42502 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.981267605299237259145031309138, −7.84016183007058400081124506286, −7.21935505245924317746793034233, −6.86562297333964146479448047341, −6.68814616729191600085892820071, −6.63382353458533584844960279022, −6.57584905763154877516153030602, −6.23727846449937061979343872709, −6.05455873304888378213257825694, −5.60923465722218931541463447817, −5.33844204460762487397124468854, −4.87466340144551585498955815265, −4.75268583486554641602332119434, −4.45622046265954078035905775448, −4.40948195333990313019834533205, −4.10166618484140994166235037567, −3.74399786939497344700652267645, −3.56652838186879732080838600023, −3.00945220165908708346934988104, −2.59911001528112120346292553136, −2.57591457004942958066563330914, −1.99797892323627015853310780207, −1.52350592846779138233653270045, −0.999023699451648926557985551450, −0.50894266224716562366475865466,
0.50894266224716562366475865466, 0.999023699451648926557985551450, 1.52350592846779138233653270045, 1.99797892323627015853310780207, 2.57591457004942958066563330914, 2.59911001528112120346292553136, 3.00945220165908708346934988104, 3.56652838186879732080838600023, 3.74399786939497344700652267645, 4.10166618484140994166235037567, 4.40948195333990313019834533205, 4.45622046265954078035905775448, 4.75268583486554641602332119434, 4.87466340144551585498955815265, 5.33844204460762487397124468854, 5.60923465722218931541463447817, 6.05455873304888378213257825694, 6.23727846449937061979343872709, 6.57584905763154877516153030602, 6.63382353458533584844960279022, 6.68814616729191600085892820071, 6.86562297333964146479448047341, 7.21935505245924317746793034233, 7.84016183007058400081124506286, 7.981267605299237259145031309138
