| L(s) = 1 | − 4·9-s − 24·17-s − 8·25-s + 52·49-s + 56·73-s − 26·81-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 96·153-s + 157-s + 163-s + 167-s − 76·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 5.82·17-s − 8/5·25-s + 52/7·49-s + 6.55·73-s − 2.88·81-s − 8·121-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 + 7.76·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 5.84·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5045398401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5045398401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + T^{2} + p^{2} T^{4} )^{4} \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + p T^{2} )^{8} \) |
| 13 | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 3 T + p T^{2} )^{8} \) |
| 23 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| 67 | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 7 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.14718359850822826139172973262, −4.06221029139059275431661223399, −4.01017026274360063816103681163, −3.96724887507630970068491393738, −3.67219907490891301998952603979, −3.58103567637207080362136103586, −3.36092327294696662444041609657, −3.30990611421250955451208655633, −2.96014391000346743989950961517, −2.92841693220948713389994166421, −2.58731939694500820612134034625, −2.51939043999563612524244591146, −2.45215125955196814113054812171, −2.36995107046449155993594136316, −2.31111772095551403053240269003, −2.30826587226868188248152410530, −1.98163288042046773655853772138, −1.91414723239338333977799859290, −1.56894531635064829727391448516, −1.42459055477061430825202347179, −1.17975443599525605456101406317, −0.857626994542693978182521476933, −0.61462210392373565737251189020, −0.41008454955637129529280685348, −0.12294204349789222863416519535,
0.12294204349789222863416519535, 0.41008454955637129529280685348, 0.61462210392373565737251189020, 0.857626994542693978182521476933, 1.17975443599525605456101406317, 1.42459055477061430825202347179, 1.56894531635064829727391448516, 1.91414723239338333977799859290, 1.98163288042046773655853772138, 2.30826587226868188248152410530, 2.31111772095551403053240269003, 2.36995107046449155993594136316, 2.45215125955196814113054812171, 2.51939043999563612524244591146, 2.58731939694500820612134034625, 2.92841693220948713389994166421, 2.96014391000346743989950961517, 3.30990611421250955451208655633, 3.36092327294696662444041609657, 3.58103567637207080362136103586, 3.67219907490891301998952603979, 3.96724887507630970068491393738, 4.01017026274360063816103681163, 4.06221029139059275431661223399, 4.14718359850822826139172973262
Plot not available for L-functions of degree greater than 10.
