| L(s) = 1 | − 8·11-s − 8·37-s − 48·49-s + 16·59-s + 24·61-s + 72·83-s − 64·97-s − 80·107-s + 48·109-s + 32·121-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 + ⋯ |
| L(s) = 1 | − 2.41·11-s − 1.31·37-s − 6.85·49-s + 2.08·59-s + 3.07·61-s + 7.90·83-s − 6.49·97-s − 7.73·107-s + 4.59·109-s + 2.90·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.766016914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.766016914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4 T^{4} - 474 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 4 T + 8 T^{2} + 28 T^{3} + 82 T^{4} + 28 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 40 T^{2} + 786 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 292 T^{4} - 25242 T^{8} + 292 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 - 964 T^{4} + 566886 T^{8} - 964 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 40 T^{2} + 594 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 8 T^{2} + 60 T^{3} - 34 T^{4} + 60 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 40 T^{2} + 2034 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 3548 T^{4} + 6433446 T^{8} - 3548 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( 1 - 4868 T^{4} + 16478118 T^{8} - 4868 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 8 T + 32 T^{2} + 232 T^{3} - 6062 T^{4} + 232 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 10276 T^{4} + 56007654 T^{8} + 10276 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 188 T^{2} + 18726 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 168 T^{2} + 14738 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 36 T + 648 T^{2} - 8604 T^{3} + 89906 T^{4} - 8604 p T^{5} + 648 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 80 T^{2} + 15714 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.59734722044519364415966080013, −3.59446969905997036845668496088, −3.51974659857022316119987664819, −3.46787550726838133164090244903, −3.36250602441742128959896593745, −3.03195071564930416978052538245, −3.02874855670634958802701592837, −2.98021018843071462199823384124, −2.79422084790122194548060652142, −2.64521099728700623980441219407, −2.38855771135429237623776280046, −2.36077021330459916152926789421, −2.32646648207678588739285485842, −2.27977215635795330248047735685, −1.91004914900317177591082140670, −1.84728794369620149159849020389, −1.63863866758674541604397437853, −1.58079401963979288701041509070, −1.39003017421898910051203143787, −1.32940899417916214123577403598, −0.890472788556210455252287597571, −0.812091932633538880344048000815, −0.43732595319430542102202798815, −0.41397637437365393184505697432, −0.16818737103510588741101270856,
0.16818737103510588741101270856, 0.41397637437365393184505697432, 0.43732595319430542102202798815, 0.812091932633538880344048000815, 0.890472788556210455252287597571, 1.32940899417916214123577403598, 1.39003017421898910051203143787, 1.58079401963979288701041509070, 1.63863866758674541604397437853, 1.84728794369620149159849020389, 1.91004914900317177591082140670, 2.27977215635795330248047735685, 2.32646648207678588739285485842, 2.36077021330459916152926789421, 2.38855771135429237623776280046, 2.64521099728700623980441219407, 2.79422084790122194548060652142, 2.98021018843071462199823384124, 3.02874855670634958802701592837, 3.03195071564930416978052538245, 3.36250602441742128959896593745, 3.46787550726838133164090244903, 3.51974659857022316119987664819, 3.59446969905997036845668496088, 3.59734722044519364415966080013
Plot not available for L-functions of degree greater than 10.
