| L(s) = 1 | + 8·19-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 + 251-s − 256-s + 257-s + ⋯ |
| L(s) = 1 | + 8·19-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 + 251-s − 256-s + 257-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.334350675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.334350675\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 - T )^{8} \) |
| good | 3 | \( ( 1 + T^{8} )^{2} \) |
| 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 + T^{4} )^{4} \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{4} \) |
| 23 | \( ( 1 + T^{4} )^{4} \) |
| 29 | \( ( 1 + T^{2} )^{8} \) |
| 31 | \( ( 1 + T^{2} )^{8} \) |
| 37 | \( ( 1 + T^{8} )^{2} \) |
| 41 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{8} )^{2} \) |
| 59 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 61 | \( ( 1 + T^{4} )^{4} \) |
| 67 | \( ( 1 + T^{8} )^{2} \) |
| 71 | \( ( 1 + T^{2} )^{8} \) |
| 73 | \( ( 1 + T^{4} )^{4} \) |
| 79 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 + T^{2} )^{8} \) |
| 97 | \( ( 1 + T^{8} )^{2} \) |
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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
−4.01514808232187882780723276098, −3.73220990931999586839118778391, −3.70952165612971071823472358277, −3.68979287640382563508939471400, −3.59628911183967796773882820332, −3.48903225471807521387987812545, −3.33533752556147130701904857578, −3.32533611319583531106423311152, −3.27265439433177827257583767795, −2.97423700743361914369795117728, −2.80330009239656723042943288687, −2.71710528807948341605507902848, −2.68297613835421237226703120875, −2.46411117762019502850953126992, −2.45836522890461094581123297629, −2.26597779040219881941539119828, −2.01821344371936332339271269048, −1.65932673570306424922656240100, −1.64253306514119950758168321757, −1.41162766731036922469977169289, −1.40899582786016670571648708225, −1.09114076169499425814473045388, −1.01674185287284018880123620885, −0.949093783544591426102224395338, −0.65401918380539009953019028975,
0.65401918380539009953019028975, 0.949093783544591426102224395338, 1.01674185287284018880123620885, 1.09114076169499425814473045388, 1.40899582786016670571648708225, 1.41162766731036922469977169289, 1.64253306514119950758168321757, 1.65932673570306424922656240100, 2.01821344371936332339271269048, 2.26597779040219881941539119828, 2.45836522890461094581123297629, 2.46411117762019502850953126992, 2.68297613835421237226703120875, 2.71710528807948341605507902848, 2.80330009239656723042943288687, 2.97423700743361914369795117728, 3.27265439433177827257583767795, 3.32533611319583531106423311152, 3.33533752556147130701904857578, 3.48903225471807521387987812545, 3.59628911183967796773882820332, 3.68979287640382563508939471400, 3.70952165612971071823472358277, 3.73220990931999586839118778391, 4.01514808232187882780723276098
Plot not available for L-functions of degree greater than 10.
