| L(s) = 1 | − 2·2-s + 5·5-s + 4·7-s + 8·8-s − 10·10-s − 42·11-s − 20·13-s − 8·14-s − 16·16-s + 186·17-s + 118·19-s + 84·22-s − 9·23-s + 40·26-s − 120·29-s − 47·31-s − 372·34-s + 20·35-s − 524·37-s − 236·38-s + 40·40-s − 126·41-s + 178·43-s + 18·46-s − 144·47-s + 343·49-s + 1.48e3·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.447·5-s + 0.215·7-s + 0.353·8-s − 0.316·10-s − 1.15·11-s − 0.426·13-s − 0.152·14-s − 1/4·16-s + 2.65·17-s + 1.42·19-s + 0.814·22-s − 0.0815·23-s + 0.301·26-s − 0.768·29-s − 0.272·31-s − 1.87·34-s + 0.0965·35-s − 2.32·37-s − 1.00·38-s + 0.158·40-s − 0.479·41-s + 0.631·43-s + 0.0576·46-s − 0.446·47-s + 49-s + 3.84·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.541815359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541815359\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T - 327 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 42 T + 433 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T - 1797 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 93 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 59 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T - 12086 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 120 T - 9989 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 47 T - 27582 T^{2} + 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 262 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 126 T - 53045 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 178 T - 47823 T^{2} - 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 144 T - 83087 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 741 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 444 T - 8243 T^{2} - 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 221 T - 178140 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 538 T - 11319 T^{2} - 538 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 690 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1126 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 665 T - 50814 T^{2} + 665 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 75 T - 566162 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1086 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1544 T + 1471263 T^{2} + 1544 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.15964445527995928547063944557, −9.668804485263512851769782233707, −9.412087943047277647548016605670, −8.705959307772179576823195293378, −8.369767862771580510344024909728, −8.076580631829488279265429604700, −7.37467071759315143195909536313, −7.13788510808669344188603481550, −7.11709146750802449144862451630, −5.76520253857933826285823931030, −5.69291646384217765238515103626, −5.18778693460000257478791995435, −5.13374816996090467860421100009, −3.86112957081186773296972175141, −3.78391039377971092125854176048, −2.85382063271838268532206752622, −2.56800762964511819174453081283, −1.59018223695680602645219575109, −1.19479600255922974365978127340, −0.41053143475731340760894048316,
0.41053143475731340760894048316, 1.19479600255922974365978127340, 1.59018223695680602645219575109, 2.56800762964511819174453081283, 2.85382063271838268532206752622, 3.78391039377971092125854176048, 3.86112957081186773296972175141, 5.13374816996090467860421100009, 5.18778693460000257478791995435, 5.69291646384217765238515103626, 5.76520253857933826285823931030, 7.11709146750802449144862451630, 7.13788510808669344188603481550, 7.37467071759315143195909536313, 8.076580631829488279265429604700, 8.369767862771580510344024909728, 8.705959307772179576823195293378, 9.412087943047277647548016605670, 9.668804485263512851769782233707, 10.15964445527995928547063944557
