| L(s) = 1 | − 1.45e3·3-s + 4.07e4·5-s + 2.10e4·7-s + 1.59e6·9-s − 6.72e5·11-s + 1.75e7·13-s − 5.93e7·15-s + 8.38e7·17-s − 2.56e8·19-s − 3.06e7·21-s − 8.59e8·23-s − 9.07e8·25-s − 1.54e9·27-s − 4.72e9·29-s + 5.98e9·31-s + 9.80e8·33-s + 8.55e8·35-s + 2.74e10·37-s − 2.55e10·39-s + 1.52e10·41-s + 1.13e10·43-s + 6.49e10·45-s + 6.90e10·47-s + 1.82e10·49-s − 1.22e11·51-s − 2.26e11·53-s − 2.73e10·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.16·5-s + 0.0674·7-s + 9-s − 0.114·11-s + 1.00·13-s − 1.34·15-s + 0.842·17-s − 1.24·19-s − 0.0779·21-s − 1.21·23-s − 0.743·25-s − 0.769·27-s − 1.47·29-s + 1.21·31-s + 0.132·33-s + 0.0786·35-s + 1.75·37-s − 1.16·39-s + 0.501·41-s + 0.272·43-s + 1.16·45-s + 0.934·47-s + 0.187·49-s − 0.972·51-s − 1.40·53-s − 0.133·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.358191836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.358191836\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 40716 T + 102620542 p^{2} T^{2} - 40716 p^{13} T^{3} + p^{26} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 21008 T - 2538590718 p T^{2} - 21008 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 61128 p T + 375287075254 p^{2} T^{2} + 61128 p^{14} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 17532604 T + 539517124797054 T^{2} - 17532604 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 83838564 T + 21472050522493798 T^{2} - 83838564 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 256293544 T + 90096185084470998 T^{2} + 256293544 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 859581936 T + 950532293946211246 T^{2} + 859581936 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4728475332 T + 25449630283285666078 T^{2} + 4728475332 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5982551648 T + 36024474609054776382 T^{2} - 5982551648 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 27411194092 T + \)\(66\!\cdots\!74\)\( T^{2} - 27411194092 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15258974292 T + \)\(17\!\cdots\!82\)\( T^{2} - 15258974292 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11314499240 T + \)\(17\!\cdots\!50\)\( T^{2} - 11314499240 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 69035142240 T + \)\(36\!\cdots\!10\)\( T^{2} - 69035142240 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 226336894164 T + \)\(34\!\cdots\!66\)\( T^{2} + 226336894164 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 927820824264 T + \)\(42\!\cdots\!38\)\( T^{2} - 927820824264 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 179395461340 T + \)\(28\!\cdots\!98\)\( T^{2} - 179395461340 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 698315061176 T + \)\(56\!\cdots\!18\)\( T^{2} - 698315061176 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 784458549936 T + \)\(20\!\cdots\!46\)\( T^{2} - 784458549936 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1857400245076 T + \)\(33\!\cdots\!66\)\( T^{2} - 1857400245076 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 714025470080 T + \)\(94\!\cdots\!78\)\( T^{2} - 714025470080 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4574293917912 T + \)\(22\!\cdots\!18\)\( T^{2} - 4574293917912 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3270178701684 T + \)\(37\!\cdots\!58\)\( T^{2} - 3270178701684 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9874926156476 T + \)\(13\!\cdots\!98\)\( T^{2} + 9874926156476 p^{13} T^{3} + p^{26} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.86928391774597495948634265925, −12.81986112371564093178509637283, −11.80490324880820213979317235407, −11.46071996007228786924251238626, −10.73070031464372758304221194442, −10.31882524248768349285674351083, −9.623800530201164639633538920318, −9.321332434343991757595260221964, −8.068577097901991194514111609116, −7.88119569408220755354409103092, −6.50549170490810188729060661382, −6.40743621889110599382331240591, −5.53689700675781728213468724811, −5.45475247685420354201800643849, −4.13313491338133497677603207423, −3.87645735237219518868065811073, −2.41778227089050283975623636084, −1.95384478394425060703331039107, −1.10403416248744024729099368928, −0.49043247643839180838802191318,
0.49043247643839180838802191318, 1.10403416248744024729099368928, 1.95384478394425060703331039107, 2.41778227089050283975623636084, 3.87645735237219518868065811073, 4.13313491338133497677603207423, 5.45475247685420354201800643849, 5.53689700675781728213468724811, 6.40743621889110599382331240591, 6.50549170490810188729060661382, 7.88119569408220755354409103092, 8.068577097901991194514111609116, 9.321332434343991757595260221964, 9.623800530201164639633538920318, 10.31882524248768349285674351083, 10.73070031464372758304221194442, 11.46071996007228786924251238626, 11.80490324880820213979317235407, 12.81986112371564093178509637283, 12.86928391774597495948634265925
