| L(s) = 1 | + 1.02e3·17-s + 1.08e3·25-s + 2.18e3·41-s + 2.43e3·49-s − 3.48e4·73-s − 1.31e4·81-s + 2.04e3·89-s + 3.07e3·97-s − 2.22e4·113-s − 1.43e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.13e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 3.54·17-s + 1.73·25-s + 1.29·41-s + 1.01·49-s − 6.53·73-s − 2·81-s + 0.258·89-s + 0.326·97-s − 1.74·113-s − 0.979·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 2.14·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.778759594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.778759594\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{8} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 542 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 174 p T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 7168 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 30686 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 256 T + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 215040 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 527426 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 294562 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1832706 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 3531490 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 546 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 5476352 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 8325762 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 15564130 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 18529280 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 6400930 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 38728704 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 24928962 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8704 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 67449218 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 75758592 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 512 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 768 T + p^{4} T^{2} )^{4} \) |
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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
−7.24604254441950442878997783397, −7.13314288553093183418306461744, −6.82965956203626374012152151645, −6.53533675115110283992094943642, −6.07003962968363609994662440756, −5.97435896881929439722110111454, −5.73855656940360589395242951041, −5.52403795121949815304737608581, −5.35836752191603072713912979195, −5.10504254309292755261234382268, −4.55107047252074454035121976080, −4.39183344459450695620937885112, −4.38912425346711502969210962858, −3.73932521637579623815997851854, −3.62042691944532924287986830680, −3.26983639516457411819423347163, −2.82481409416196266658430201260, −2.75744171769716654751044390332, −2.73834088787519785952970601607, −1.93729418514698736791890193452, −1.46404414766930427954847262954, −1.27745816229973963939173049789, −1.16057206813097296598571534314, −0.62188941132404577887454368638, −0.31445776728347072877485766005,
0.31445776728347072877485766005, 0.62188941132404577887454368638, 1.16057206813097296598571534314, 1.27745816229973963939173049789, 1.46404414766930427954847262954, 1.93729418514698736791890193452, 2.73834088787519785952970601607, 2.75744171769716654751044390332, 2.82481409416196266658430201260, 3.26983639516457411819423347163, 3.62042691944532924287986830680, 3.73932521637579623815997851854, 4.38912425346711502969210962858, 4.39183344459450695620937885112, 4.55107047252074454035121976080, 5.10504254309292755261234382268, 5.35836752191603072713912979195, 5.52403795121949815304737608581, 5.73855656940360589395242951041, 5.97435896881929439722110111454, 6.07003962968363609994662440756, 6.53533675115110283992094943642, 6.82965956203626374012152151645, 7.13314288553093183418306461744, 7.24604254441950442878997783397
