L(s) = 1 | + 20·25-s + 56·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 13·256-s + ⋯ |
L(s) = 1 | + 4·25-s + 8·49-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + 0.812·256-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.103624925\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.103624925\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( ( 1 - p T^{2} )^{4} \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 13 T^{8} + p^{8} T^{16} \) |
| 3 | \( 1 + 142 T^{8} + p^{8} T^{16} \) |
| 7 | \( ( 1 - p T^{2} )^{8} \) |
| 11 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 + 52622 T^{8} + p^{8} T^{16} \) |
| 17 | \( ( 1 - p T^{2} )^{8} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( 1 - 3237298 T^{8} + p^{8} T^{16} \) |
| 41 | \( ( 1 + p T^{2} )^{8} \) |
| 43 | \( ( 1 - p T^{2} )^{8} \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( 1 + 15154382 T^{8} + p^{8} T^{16} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - 7138 T^{4} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 3364622 T^{8} + p^{8} T^{16} \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 + p T^{2} )^{8} \) |
| 83 | \( ( 1 - p T^{2} )^{8} \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( 1 - 60577618 T^{8} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.95711360073062450895337654813, −3.67087834470992182704411540939, −3.55738768756499499497728199727, −3.55598562744537075653650478976, −3.52027701952687791433166308846, −3.33146102065280175589905290480, −3.04397293363790477363660306673, −3.04115792118313243600773856010, −2.82854609300446008671748082859, −2.68882304434766406854133344512, −2.65522743083767472106731319547, −2.61068377107659626706011855597, −2.37190348813924003405344479745, −2.22780394731086744065333156101, −2.05666627198079070150501027710, −1.94039344724753856668005228458, −1.80680952843158242661251032438, −1.71619025018028218358944602134, −1.22649099254439621590231990928, −1.15261825081475647673729503567, −0.932849486335523003189705802122, −0.930290577331421066070869268059, −0.75997196959084097768894681056, −0.63767766363901653730748255257, −0.19794344001578377881264968084,
0.19794344001578377881264968084, 0.63767766363901653730748255257, 0.75997196959084097768894681056, 0.930290577331421066070869268059, 0.932849486335523003189705802122, 1.15261825081475647673729503567, 1.22649099254439621590231990928, 1.71619025018028218358944602134, 1.80680952843158242661251032438, 1.94039344724753856668005228458, 2.05666627198079070150501027710, 2.22780394731086744065333156101, 2.37190348813924003405344479745, 2.61068377107659626706011855597, 2.65522743083767472106731319547, 2.68882304434766406854133344512, 2.82854609300446008671748082859, 3.04115792118313243600773856010, 3.04397293363790477363660306673, 3.33146102065280175589905290480, 3.52027701952687791433166308846, 3.55598562744537075653650478976, 3.55738768756499499497728199727, 3.67087834470992182704411540939, 3.95711360073062450895337654813
Plot not available for L-functions of degree greater than 10.