| L(s) = 1 | + 486·9-s − 1.15e4·17-s − 1.43e4·25-s + 7.49e4·41-s + 3.24e5·49-s + 2.51e6·73-s + 1.77e5·81-s − 5.50e6·89-s + 1.95e6·97-s − 6.32e6·113-s − 1.13e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 5.63e6·153-s + 157-s + 163-s + 167-s + 2.71e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2.35·17-s − 0.917·25-s + 1.08·41-s + 2.75·49-s + 6.45·73-s + 1/3·81-s − 7.80·89-s + 2.14·97-s − 4.38·113-s − 0.640·121-s − 1.57·153-s + 0.563·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.325539905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.325539905\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 7166 T^{2} + p^{12} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 162290 T^{2} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 567074 T^{2} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 1359218 T^{2} + p^{12} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2898 T + p^{6} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 1743550 T^{2} + p^{12} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 293443490 T^{2} + p^{12} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 976603426 T^{2} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1727677010 T^{2} + p^{12} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 4983392594 T^{2} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 18738 T + p^{6} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 5648607070 T^{2} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 1324300030 T^{2} + p^{12} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 23524698562 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 26964818206 T^{2} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8136149902 T^{2} + p^{12} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 78875702786 T^{2} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 79263457634 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8606 p T + p^{6} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 400981984274 T^{2} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 526419436610 T^{2} + p^{12} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 1375074 T + p^{6} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 489710 T + p^{6} T^{2} )^{4} \) |
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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
−7.09043334192746213042732190944, −6.96611170853890438030759441486, −6.74290217924314945939109168217, −6.45794921491943076880096618464, −6.12679983564161528881908034032, −6.09200837984116699002760829127, −5.46593869072315604433969721156, −5.40609899904537487430425394022, −5.20248663862540180314850241707, −4.90992198146446574304214192404, −4.28314039892719423353218960884, −4.21204837772251925100569426271, −4.17506872885539156188605915225, −3.88583530045502840240003720090, −3.58851203622710696686237322064, −2.98500428587598290081915424481, −2.76460912106516714180348634485, −2.48133115201043464525156429490, −2.13810960387850130860096271744, −2.01966427262219881908699252172, −1.57187398046978677362870380447, −1.24665184994561214639361025379, −0.67177025891582614369999627356, −0.66696729893060139318968681303, −0.18294508809648738156515284914,
0.18294508809648738156515284914, 0.66696729893060139318968681303, 0.67177025891582614369999627356, 1.24665184994561214639361025379, 1.57187398046978677362870380447, 2.01966427262219881908699252172, 2.13810960387850130860096271744, 2.48133115201043464525156429490, 2.76460912106516714180348634485, 2.98500428587598290081915424481, 3.58851203622710696686237322064, 3.88583530045502840240003720090, 4.17506872885539156188605915225, 4.21204837772251925100569426271, 4.28314039892719423353218960884, 4.90992198146446574304214192404, 5.20248663862540180314850241707, 5.40609899904537487430425394022, 5.46593869072315604433969721156, 6.09200837984116699002760829127, 6.12679983564161528881908034032, 6.45794921491943076880096618464, 6.74290217924314945939109168217, 6.96611170853890438030759441486, 7.09043334192746213042732190944
