| L(s) = 1 | − 4·7-s + 20·9-s + 8·11-s + 24·23-s + 104·29-s − 40·37-s + 184·43-s − 54·49-s + 152·53-s − 80·63-s − 8·67-s − 248·71-s − 32·77-s − 248·79-s + 170·81-s + 160·99-s − 328·107-s − 88·109-s − 168·113-s − 188·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 4/7·7-s + 20/9·9-s + 8/11·11-s + 1.04·23-s + 3.58·29-s − 1.08·37-s + 4.27·43-s − 1.10·49-s + 2.86·53-s − 1.26·63-s − 0.119·67-s − 3.49·71-s − 0.415·77-s − 3.13·79-s + 2.09·81-s + 1.61·99-s − 3.06·107-s − 0.807·109-s − 1.48·113-s − 1.55·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.234693764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.234693764\) |
| \(L(2)\) |
|
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 | $C_4$ | \( 1 + 4 T + 10 p T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 - 20 T^{2} + 230 T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 118 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 212 T^{2} + 14566 T^{4} - 212 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 900 T^{2} + 361350 T^{4} - 900 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 404 T^{2} + 284518 T^{4} - 404 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 12 T + 582 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 52 T + 2230 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 3332 T^{2} + 4589830 T^{4} - 3332 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 20 T + 1686 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 1284 T^{2} + 1246278 T^{4} - 1284 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 1412 T^{2} - 3512954 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 76 T + 6550 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 10324 T^{2} + 48390054 T^{4} - 10324 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 7252 T^{2} + 27026790 T^{4} - 7252 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 4 T + 8470 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 124 T + 13638 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 18180 T^{2} + 138195270 T^{4} - 18180 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 124 T + 15526 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 16404 T^{2} + 162018918 T^{4} - 16404 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 25476 T^{2} + 278131398 T^{4} - 25476 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 4356 T^{2} + 109417734 T^{4} - 4356 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.73147254683373185218011639604, −6.38714202434317019682789593627, −6.32807727360424300405485458845, −6.00721145414190127826173327411, −5.84242072649103473628983316280, −5.43031861541026629640625546445, −5.42567156523671164059741680898, −5.00254489056084325044091129373, −4.70930535060190227161086368509, −4.52698875497904656360240948490, −4.43319182169346652979800617441, −4.14948623310363938593532683443, −3.88082322373967530988316613199, −3.76741264108982031576353205882, −3.61634177916590204753174950447, −2.77454882298285702945197345052, −2.72011333868748552138940341654, −2.70205157744675835668305321612, −2.69351505566590726147405406100, −1.85124006772391774286668244385, −1.51372414344513697842913204715, −1.20244546593877988263165722333, −1.19819538181599724815877955294, −0.825567186117410620553509172948, −0.28337676694493490113256748270,
0.28337676694493490113256748270, 0.825567186117410620553509172948, 1.19819538181599724815877955294, 1.20244546593877988263165722333, 1.51372414344513697842913204715, 1.85124006772391774286668244385, 2.69351505566590726147405406100, 2.70205157744675835668305321612, 2.72011333868748552138940341654, 2.77454882298285702945197345052, 3.61634177916590204753174950447, 3.76741264108982031576353205882, 3.88082322373967530988316613199, 4.14948623310363938593532683443, 4.43319182169346652979800617441, 4.52698875497904656360240948490, 4.70930535060190227161086368509, 5.00254489056084325044091129373, 5.42567156523671164059741680898, 5.43031861541026629640625546445, 5.84242072649103473628983316280, 6.00721145414190127826173327411, 6.32807727360424300405485458845, 6.38714202434317019682789593627, 6.73147254683373185218011639604
