| L(s) = 1 | + 18·9-s + 36·29-s + 20·41-s + 50·49-s + 36·61-s + 169·81-s − 52·89-s − 68·101-s + 12·109-s + 16·121-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 | + 6·9-s + 6.68·29-s + 3.12·41-s + 50/7·49-s + 4.60·61-s + 18.7·81-s − 5.51·89-s − 6.76·101-s + 1.14·109-s + 1.45·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(148.4788423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(148.4788423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 - p^{2} T^{2} + 37 T^{4} - p^{4} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 25 T^{2} + 253 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 8 T^{2} + 238 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 318 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 78 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 8 T^{2} + 238 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 65 T^{2} + 2053 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 72 T^{2} + 3198 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 60 T^{2} + 2358 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 5 T + 77 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 9 T^{2} - 1243 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 85 T^{2} + 3693 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T^{2} + 5302 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 24 T^{2} + 6606 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 156 T^{2} + 12182 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 216 T^{2} + 21246 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 32 T^{2} + 10414 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 228 T^{2} + 24198 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 117 T^{2} + 14669 T^{4} - 117 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 13 T + 209 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 128 T^{2} + 22414 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.47280120324780030742893062310, −3.43182410906491614551828974976, −3.41702495852586601734652772615, −3.02135565474564956268086812164, −2.91911082851331427151859870547, −2.77037119646334292503654871419, −2.70220944367321993738853576689, −2.59585774011404505065622062933, −2.56983087016007383941277321899, −2.43066483551502325190904658313, −2.42100394661581418740121015500, −2.29044976993426780892005412851, −1.97765129691247100050843085234, −1.85051939187392883374763153374, −1.69201047273722483810939653880, −1.57738329273548393248130792046, −1.42773235487355223101937324935, −1.38215660771820369211985995397, −1.10157069213622129244525142039, −1.02686067417399584818603187381, −1.00101565868482520041730412843, −0.75630258947216390081934954928, −0.68297509971162117950912920572, −0.66300560973275060697483234331, −0.44520424256793308994456921974,
0.44520424256793308994456921974, 0.66300560973275060697483234331, 0.68297509971162117950912920572, 0.75630258947216390081934954928, 1.00101565868482520041730412843, 1.02686067417399584818603187381, 1.10157069213622129244525142039, 1.38215660771820369211985995397, 1.42773235487355223101937324935, 1.57738329273548393248130792046, 1.69201047273722483810939653880, 1.85051939187392883374763153374, 1.97765129691247100050843085234, 2.29044976993426780892005412851, 2.42100394661581418740121015500, 2.43066483551502325190904658313, 2.56983087016007383941277321899, 2.59585774011404505065622062933, 2.70220944367321993738853576689, 2.77037119646334292503654871419, 2.91911082851331427151859870547, 3.02135565474564956268086812164, 3.41702495852586601734652772615, 3.43182410906491614551828974976, 3.47280120324780030742893062310
Plot not available for L-functions of degree greater than 10.
