| L(s) = 1 | − 2·2-s − 3·3-s + 5·5-s + 6·6-s − 20·7-s + 8·8-s − 10·10-s + 19·11-s + 66·13-s + 40·14-s − 15·15-s − 16·16-s − 64·17-s + 141·19-s + 60·21-s − 38·22-s + 51·23-s − 24·24-s − 132·26-s + 27·27-s + 432·29-s + 30·30-s − 290·31-s − 57·33-s + 128·34-s − 100·35-s + 109·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 1.07·7-s + 0.353·8-s − 0.316·10-s + 0.520·11-s + 1.40·13-s + 0.763·14-s − 0.258·15-s − 1/4·16-s − 0.913·17-s + 1.70·19-s + 0.623·21-s − 0.368·22-s + 0.462·23-s − 0.204·24-s − 0.995·26-s + 0.192·27-s + 2.76·29-s + 0.182·30-s − 1.68·31-s − 0.300·33-s + 0.645·34-s − 0.482·35-s + 0.484·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.396236567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396236567\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + 20 T + p^{3} T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 19 T - 970 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 33 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 64 T - 817 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 141 T + 13022 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 51 T - 9566 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 216 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 290 T + 54309 T^{2} + 290 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 109 T - 38772 T^{2} - 109 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 457 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 184 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 313 T - 5854 T^{2} + 313 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 319 T - 47116 T^{2} - 319 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 44 T - 203443 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 368 T - 91557 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 216 T - 254107 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 314 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 602 T - 26613 T^{2} + 602 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 112 T - 480495 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 712 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1018 T + 331355 T^{2} - 1018 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 584 T + p^{3} T^{2} )^{2} \) |
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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
−11.88482972501755799620777103103, −11.78354646395459530479576449239, −10.92648469275022890655237471865, −10.74665222264693807860038382359, −10.29967395080665325931608289701, −9.397986649916623518187596551605, −9.302513734244584367872384755259, −9.046724968309015254554390057618, −8.195705564233391052947498429928, −7.70660123411610248959588255537, −6.85703261616326483296820177965, −6.65728881858090487086155888246, −5.80456593914683726292325215723, −5.75967358696947002733458399775, −4.66161492399431722666381766915, −4.06898224956234262976848068582, −3.23524799729101317138375595009, −2.53156898488651107387105562091, −1.19078271404734542163613136837, −0.71101474296721254373475947982,
0.71101474296721254373475947982, 1.19078271404734542163613136837, 2.53156898488651107387105562091, 3.23524799729101317138375595009, 4.06898224956234262976848068582, 4.66161492399431722666381766915, 5.75967358696947002733458399775, 5.80456593914683726292325215723, 6.65728881858090487086155888246, 6.85703261616326483296820177965, 7.70660123411610248959588255537, 8.195705564233391052947498429928, 9.046724968309015254554390057618, 9.302513734244584367872384755259, 9.397986649916623518187596551605, 10.29967395080665325931608289701, 10.74665222264693807860038382359, 10.92648469275022890655237471865, 11.78354646395459530479576449239, 11.88482972501755799620777103103
