| L(s) = 1 | − 812·13-s − 340·25-s + 44·37-s + 6.95e3·49-s − 7.92e3·61-s − 1.66e4·73-s + 6.73e4·97-s − 6.05e3·109-s + 244·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.97e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4.80·13-s − 0.543·25-s + 0.0321·37-s + 2.89·49-s − 2.12·61-s − 3.11·73-s + 7.16·97-s − 0.509·109-s + 0.0166·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 10.4·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5594372697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5594372697\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 34 p T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 13 p T + p^{4} T^{2} )^{2}( 1 + 13 p T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 203 T + p^{4} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 165962 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 74639 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 169318 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 567842 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1819394 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p^{4} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 4301758 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2688674 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 8330522 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 13875842 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 11486278 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 1981 T + p^{4} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 4283567 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8716322 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4151 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 43606679 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 89201282 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 122217482 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16849 T + p^{4} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41559703429318283499032515218, −7.21995065396024180142066781257, −7.17295889345570528668253972227, −6.92805738065015183197399167903, −6.27648049597898456328950183615, −6.15216783167223521069462741955, −5.97257000963473345690361105556, −5.68443496152248901587180940863, −5.23366743487016846241560997444, −5.08332070068834069878440585607, −4.75407376734820470238547395198, −4.67312204271067192599726320321, −4.54515779215750918476418258133, −3.95386230885675001839256315221, −3.88257456758766355350664617953, −3.27343230998776762519853338098, −2.88546243351252850992365638207, −2.87137281703970113730281874929, −2.33954815255746232005334410822, −2.15900962655740978420782516644, −2.03499406528663238396203562497, −1.46859068584831669184279285425, −0.873199308694316866444212090103, −0.47565953355469904307169312261, −0.14440865480173824066576081295,
0.14440865480173824066576081295, 0.47565953355469904307169312261, 0.873199308694316866444212090103, 1.46859068584831669184279285425, 2.03499406528663238396203562497, 2.15900962655740978420782516644, 2.33954815255746232005334410822, 2.87137281703970113730281874929, 2.88546243351252850992365638207, 3.27343230998776762519853338098, 3.88257456758766355350664617953, 3.95386230885675001839256315221, 4.54515779215750918476418258133, 4.67312204271067192599726320321, 4.75407376734820470238547395198, 5.08332070068834069878440585607, 5.23366743487016846241560997444, 5.68443496152248901587180940863, 5.97257000963473345690361105556, 6.15216783167223521069462741955, 6.27648049597898456328950183615, 6.92805738065015183197399167903, 7.17295889345570528668253972227, 7.21995065396024180142066781257, 7.41559703429318283499032515218
