| L(s) = 1 | + 8·11-s − 8·41-s + 18·81-s − 52·121-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 + ⋯ |
| L(s) = 1 | + 2.41·11-s − 1.24·41-s + 2·81-s − 4.72·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.125091201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.125091201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 - T + p T^{2} )^{8} \) |
| 13 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 - 529 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | \( ( 1 + T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + 1778 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 1058 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4942 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 2471 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 9791 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 12791 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 2498 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−4.47329917886907985487472376669, −4.22231061131482972524554459087, −4.11961975531878262143622148832, −4.07829322438949224997724746420, −4.04675724216499875093695814488, −3.57009106363017286576727386555, −3.54566180329874361724599407920, −3.48926344481506577688212251429, −3.45714594266378735630582257357, −3.39664657143676848463732176565, −3.11788211678711047182717281385, −2.88244913855540096920653180934, −2.74365551243240221136520925054, −2.47423065662552748946728207901, −2.41438942099799612230069262479, −2.22142348442907391212073185960, −2.20760173468704494485445778690, −1.81546517934524829516181440016, −1.59174000061856329096513502089, −1.47404084694181229385956359412, −1.46668707215076335047911780158, −1.12070698651804185382596363228, −0.874179826190052999047873057620, −0.68278634439959554470537327972, −0.17895965083878145492100681923,
0.17895965083878145492100681923, 0.68278634439959554470537327972, 0.874179826190052999047873057620, 1.12070698651804185382596363228, 1.46668707215076335047911780158, 1.47404084694181229385956359412, 1.59174000061856329096513502089, 1.81546517934524829516181440016, 2.20760173468704494485445778690, 2.22142348442907391212073185960, 2.41438942099799612230069262479, 2.47423065662552748946728207901, 2.74365551243240221136520925054, 2.88244913855540096920653180934, 3.11788211678711047182717281385, 3.39664657143676848463732176565, 3.45714594266378735630582257357, 3.48926344481506577688212251429, 3.54566180329874361724599407920, 3.57009106363017286576727386555, 4.04675724216499875093695814488, 4.07829322438949224997724746420, 4.11961975531878262143622148832, 4.22231061131482972524554459087, 4.47329917886907985487472376669
Plot not available for L-functions of degree greater than 10.
