| L(s) = 1 | + 8·9-s + 16·11-s − 24·25-s + 16·43-s + 4·49-s − 80·67-s + 20·81-s + 128·99-s − 48·107-s + 48·113-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 8/3·9-s + 4.82·11-s − 4.79·25-s + 2.43·43-s + 4/7·49-s − 9.77·67-s + 20/9·81-s + 12.8·99-s − 4.64·107-s + 4.51·113-s + 5.09·121-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 − 1.84·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287642537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287642537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 - 4 T^{2} + 14 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 12 T^{2} + 78 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 13 | \( ( 1 + 12 T^{2} + 366 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 - 36 T^{2} + 1038 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 44 T^{2} + 1654 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 4 T^{2} + 3238 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 132 T^{2} + 10926 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 236 T^{2} + 21358 T^{4} + 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 20 T + 202 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 196 T^{2} + 18534 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 132 T^{2} + 8742 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 100 T^{2} + 11782 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 196 T^{2} + 19150 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 324 T^{2} + 41958 T^{4} - 324 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 68 T^{2} + 19462 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.92948778803145440600643660638, −3.76595324397513378230019649764, −3.72957566197255733811498846016, −3.72682228352386255047476290772, −3.62125528573127732887122831225, −3.22345749764427847758232998809, −3.19868257939613069899895185580, −3.13530677215445032150611345753, −3.01407803785811807668954918116, −2.75648476195064519890219266436, −2.55902134233814876576587996309, −2.47277456905559404410639601545, −2.17746145393731217744467431165, −2.17255184519555672134414334307, −2.05289748578681073416708548913, −1.74501953913121725707709187187, −1.57743717931795837972402843320, −1.49130820445570323355745148828, −1.42720733649208014820562084030, −1.36775382181584338458591232190, −1.28921093302253120324720489492, −1.04734685050917771836805748544, −0.889338992688817605774760240132, −0.35483099364878232545844217443, −0.093177152241751159389904533947,
0.093177152241751159389904533947, 0.35483099364878232545844217443, 0.889338992688817605774760240132, 1.04734685050917771836805748544, 1.28921093302253120324720489492, 1.36775382181584338458591232190, 1.42720733649208014820562084030, 1.49130820445570323355745148828, 1.57743717931795837972402843320, 1.74501953913121725707709187187, 2.05289748578681073416708548913, 2.17255184519555672134414334307, 2.17746145393731217744467431165, 2.47277456905559404410639601545, 2.55902134233814876576587996309, 2.75648476195064519890219266436, 3.01407803785811807668954918116, 3.13530677215445032150611345753, 3.19868257939613069899895185580, 3.22345749764427847758232998809, 3.62125528573127732887122831225, 3.72682228352386255047476290772, 3.72957566197255733811498846016, 3.76595324397513378230019649764, 3.92948778803145440600643660638
Plot not available for L-functions of degree greater than 10.
