| L(s) = 1 | − 1.51·2-s + 0.296·4-s − 5-s + 7-s + 2.58·8-s + 1.51·10-s − 2.21·11-s + 1.18·13-s − 1.51·14-s − 4.50·16-s − 3.39·17-s + 7.61·19-s − 0.296·20-s + 3.36·22-s − 3.39·23-s + 25-s − 1.80·26-s + 0.296·28-s − 1.40·29-s − 4.42·31-s + 1.66·32-s + 5.14·34-s − 35-s + 5.03·37-s − 11.5·38-s − 2.58·40-s + 7.80·41-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.148·4-s − 0.447·5-s + 0.377·7-s + 0.912·8-s + 0.479·10-s − 0.669·11-s + 0.329·13-s − 0.404·14-s − 1.12·16-s − 0.823·17-s + 1.74·19-s − 0.0662·20-s + 0.716·22-s − 0.707·23-s + 0.200·25-s − 0.353·26-s + 0.0559·28-s − 0.261·29-s − 0.794·31-s + 0.293·32-s + 0.881·34-s − 0.169·35-s + 0.827·37-s − 1.87·38-s − 0.408·40-s + 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7185204818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7185204818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 1.51T + 2T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 7.61T + 19T^{2} \) |
| 23 | \( 1 + 3.39T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 - 5.03T + 37T^{2} \) |
| 41 | \( 1 - 7.80T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 - 7.86T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 3.03T + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 3.18T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 7.37T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.852518434597074777781715609392, −9.235283016143865770102590406281, −8.359646667281446515959058629525, −7.71598207677455054508658507378, −7.10446625739659078882182656826, −5.71047682058079076764186071235, −4.74945753477362831722670801586, −3.73475127956385771274174590578, −2.23701096439255923637237990751, −0.794588120318086899729737301335,
0.794588120318086899729737301335, 2.23701096439255923637237990751, 3.73475127956385771274174590578, 4.74945753477362831722670801586, 5.71047682058079076764186071235, 7.10446625739659078882182656826, 7.71598207677455054508658507378, 8.359646667281446515959058629525, 9.235283016143865770102590406281, 9.852518434597074777781715609392
