| L(s) = 1 | + 7·4-s + 33·16-s − 50·25-s − 196·49-s + 472·61-s + 119·64-s − 350·100-s − 88·109-s + 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 1.37e3·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 7/4·4-s + 2.06·16-s − 2·25-s − 4·49-s + 7.73·61-s + 1.85·64-s − 7/2·100-s − 0.807·109-s + 4·121-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 + 0.00613·163-s + 0.00598·167-s + 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s − 7·196-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.926898889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.926898889\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 7 T^{2} + p^{4} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 382 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 4222 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1778 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 118 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 98 T + p^{2} T^{2} )^{2}( 1 + 98 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 9938 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
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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
−9.170748296611558363720919005926, −8.577556139829743073259236462164, −8.219653089388856729165542036522, −8.191772253089824194086351723127, −8.099139767922506522851436855834, −7.71511614337214403637906438029, −7.19612007305331832428950264862, −7.07039562035601777884115027117, −6.81560877430000993371432261209, −6.62451444504420816018397345711, −6.34282455026921302690943712313, −5.78794047065834910712782675606, −5.73481952696108318749166615691, −5.50147810588176706719404931623, −5.01489267642542554817514436976, −4.67387305211446898093836168124, −4.24590222009882136597292985208, −3.63575559246780927556807195483, −3.55505473700686498126572765941, −3.28431358399553533355316164406, −2.42700851535336932045025801689, −2.38899370000382688648787484083, −1.91892529519781189932154374780, −1.43094506493294664169337786972, −0.59616570877129900726359977756,
0.59616570877129900726359977756, 1.43094506493294664169337786972, 1.91892529519781189932154374780, 2.38899370000382688648787484083, 2.42700851535336932045025801689, 3.28431358399553533355316164406, 3.55505473700686498126572765941, 3.63575559246780927556807195483, 4.24590222009882136597292985208, 4.67387305211446898093836168124, 5.01489267642542554817514436976, 5.50147810588176706719404931623, 5.73481952696108318749166615691, 5.78794047065834910712782675606, 6.34282455026921302690943712313, 6.62451444504420816018397345711, 6.81560877430000993371432261209, 7.07039562035601777884115027117, 7.19612007305331832428950264862, 7.71511614337214403637906438029, 8.099139767922506522851436855834, 8.191772253089824194086351723127, 8.219653089388856729165542036522, 8.577556139829743073259236462164, 9.170748296611558363720919005926
