| L(s) = 1 | − 4·4-s + 10·16-s − 16·25-s + 8·37-s + 56·43-s − 20·64-s + 40·67-s + 40·79-s + 64·100-s − 40·109-s + 16·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 68·169-s − 224·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·4-s + 5/2·16-s − 3.19·25-s + 1.31·37-s + 8.53·43-s − 5/2·64-s + 4.88·67-s + 4.50·79-s + 32/5·100-s − 3.83·109-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.23·169-s − 17.0·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.441898752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.441898752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 34 T^{2} + 555 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 8 T^{2} - 894 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 10 T^{2} + 1299 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 7 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 - 40 T^{2} + 2226 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4 T^{2} - 4746 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 226 T^{2} + 20139 T^{4} - 226 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10 T + 87 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 28 T^{2} + 486 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 68 T^{2} + 4566 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 32 T^{2} - 7230 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 334 T^{2} + 46419 T^{4} - 334 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.79682439842408725645974194372, −3.63880931653081089649976108886, −3.50309032284361117387764734757, −3.38029203801264367902914301147, −3.25255356433290708160529303314, −3.15701509156585692395755774083, −2.94468464875930285391337968748, −2.83078959449366713392229140092, −2.69886050586456376622402975167, −2.68535881997343551267743626001, −2.31698841400402936106761776552, −2.26856594744872049951193987211, −2.17892325390047760207801638332, −2.14359464356999346487198401635, −2.01974174182680185381827974692, −1.79222205244829719042190605170, −1.67634309711949696381338865944, −1.54159060301672593619450066375, −0.983601803966360018328431051460, −0.974296306875470715659204744582, −0.945017880390302524310348376166, −0.67743295333349314608845653816, −0.60200044865039844948310046568, −0.59728340232999986154178215684, −0.25908327105876428669170023508,
0.25908327105876428669170023508, 0.59728340232999986154178215684, 0.60200044865039844948310046568, 0.67743295333349314608845653816, 0.945017880390302524310348376166, 0.974296306875470715659204744582, 0.983601803966360018328431051460, 1.54159060301672593619450066375, 1.67634309711949696381338865944, 1.79222205244829719042190605170, 2.01974174182680185381827974692, 2.14359464356999346487198401635, 2.17892325390047760207801638332, 2.26856594744872049951193987211, 2.31698841400402936106761776552, 2.68535881997343551267743626001, 2.69886050586456376622402975167, 2.83078959449366713392229140092, 2.94468464875930285391337968748, 3.15701509156585692395755774083, 3.25255356433290708160529303314, 3.38029203801264367902914301147, 3.50309032284361117387764734757, 3.63880931653081089649976108886, 3.79682439842408725645974194372
Plot not available for L-functions of degree greater than 10.
