| L(s) = 1 | − 24·11-s + 16-s − 40·31-s − 60·41-s + 52·61-s − 96·101-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 24·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 7.23·11-s + 1/4·16-s − 7.18·31-s − 9.37·41-s + 6.65·61-s − 9.55·101-s + 24.3·121-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 − 1.80·176-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1466198547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1466198547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 94 T^{4} + p^{4} T^{8} )( 1 + 71 T^{4} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 142 T^{4} - 8397 T^{8} + 142 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 - 311 T^{4} - 183120 T^{8} - 311 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 17 T^{2} - 552 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 1106 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 23 T^{4} + p^{4} T^{8} )( 1 + 3191 T^{4} + p^{4} T^{8} ) \) |
| 47 | \( 1 - 2807 T^{4} + 2999568 T^{8} - 2807 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + T + p T^{2} )^{4} \) |
| 67 | \( ( 1 + 2903 T^{4} + p^{4} T^{8} )( 1 + 5906 T^{4} + p^{4} T^{8} ) \) |
| 71 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 8542 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2}( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | \( 1 - 10871 T^{4} + 70720320 T^{8} - 10871 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 175 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 2498 T^{4} - 82289277 T^{8} - 2498 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−4.03081496636584565603642656209, −3.97634444321583862926620474088, −3.74890591802656943055662642400, −3.65077989707617198847631747076, −3.62765131724331748786732799590, −3.37141523751221387611495232713, −3.17209078512230818504990629488, −3.13670357996030676827160867778, −3.09375696866105144897913641469, −2.86987175340067480927822931456, −2.83155928471805643282997268903, −2.59942965380390420528307941036, −2.56229589224577831168919157328, −2.28514483977314250955482229794, −2.09432160616555126721282508934, −2.04357741884569635898848372414, −1.85681723996106518761736841303, −1.83129739823083145923337201448, −1.73507812412741950231261253609, −1.45949974125347788785392875277, −1.33698929090662285058329202715, −0.68446412509463942769998929382, −0.31281253003120742514611657326, −0.29580562985145656932893037630, −0.15878361258769678038629898561,
0.15878361258769678038629898561, 0.29580562985145656932893037630, 0.31281253003120742514611657326, 0.68446412509463942769998929382, 1.33698929090662285058329202715, 1.45949974125347788785392875277, 1.73507812412741950231261253609, 1.83129739823083145923337201448, 1.85681723996106518761736841303, 2.04357741884569635898848372414, 2.09432160616555126721282508934, 2.28514483977314250955482229794, 2.56229589224577831168919157328, 2.59942965380390420528307941036, 2.83155928471805643282997268903, 2.86987175340067480927822931456, 3.09375696866105144897913641469, 3.13670357996030676827160867778, 3.17209078512230818504990629488, 3.37141523751221387611495232713, 3.62765131724331748786732799590, 3.65077989707617198847631747076, 3.74890591802656943055662642400, 3.97634444321583862926620474088, 4.03081496636584565603642656209
Plot not available for L-functions of degree greater than 10.
