| L(s) = 1 | + 24·25-s − 48·83-s + 48·109-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 + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 24/5·25-s − 5.26·83-s + 4.59·109-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05622925329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05622925329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 146 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 60 T^{2} + 1466 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 28 T^{2} + 486 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 8 T^{2} - 654 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 76 T^{2} + 2934 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 108 T^{2} + 6170 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 76 T^{2} + 3414 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 168 T^{2} + 13730 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 100 T^{2} + 4566 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 216 T^{2} + 21554 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T^{2} + 5574 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 204 T^{2} + 25946 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 264 T^{2} + 34514 T^{4} + 264 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.13827856155495983722693542701, −3.01067195711220464809800509353, −2.92651226854356251685722648879, −2.85467524626166116162138068512, −2.82411115687996380536055747182, −2.73922976459734576863842256226, −2.68667243563036526387255894877, −2.56167518332529566576323646671, −2.42164225960277285780985688456, −2.33021492689230113163835176177, −2.01778759725552839825193247644, −1.92304012806657679482380016743, −1.76100641620780501619224831472, −1.73116753820968324721342880818, −1.56963965087870548686477305623, −1.56892806406112442696347586336, −1.53064303693446841352460777451, −1.05318654604059890849531074013, −0.944977645826937015969475996629, −0.935548871579024577715958377083, −0.922984408085791023709192296811, −0.78718699878884859166343780270, −0.59728318824190725357734001025, −0.11365654303248221709201379876, −0.02950389597638807702754808242,
0.02950389597638807702754808242, 0.11365654303248221709201379876, 0.59728318824190725357734001025, 0.78718699878884859166343780270, 0.922984408085791023709192296811, 0.935548871579024577715958377083, 0.944977645826937015969475996629, 1.05318654604059890849531074013, 1.53064303693446841352460777451, 1.56892806406112442696347586336, 1.56963965087870548686477305623, 1.73116753820968324721342880818, 1.76100641620780501619224831472, 1.92304012806657679482380016743, 2.01778759725552839825193247644, 2.33021492689230113163835176177, 2.42164225960277285780985688456, 2.56167518332529566576323646671, 2.68667243563036526387255894877, 2.73922976459734576863842256226, 2.82411115687996380536055747182, 2.85467524626166116162138068512, 2.92651226854356251685722648879, 3.01067195711220464809800509353, 3.13827856155495983722693542701
Plot not available for L-functions of degree greater than 10.
