| L(s) = 1 | − 1.78·3-s + (0.615 − 2.14i)5-s − 2.64·7-s + 0.171·9-s − 0.737i·13-s + (−1.09 + 3.82i)15-s + 5.43i·19-s + 4.71·21-s − 7.48·23-s + (−4.24 − 2.64i)25-s + 5.03·27-s + (−1.62 + 5.68i)35-s + 1.31i·39-s + (0.105 − 0.368i)45-s + 7.00·49-s + ⋯ |
| L(s) = 1 | − 1.02·3-s + (0.275 − 0.961i)5-s − 0.999·7-s + 0.0571·9-s − 0.204i·13-s + (−0.282 + 0.988i)15-s + 1.24i·19-s + 1.02·21-s − 1.56·23-s + (−0.848 − 0.529i)25-s + 0.969·27-s + (−0.275 + 0.961i)35-s + 0.210i·39-s + (0.0157 − 0.0549i)45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6560232130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6560232130\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.615 + 2.14i)T \) |
| 7 | \( 1 + 2.64T \) |
| good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.737iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.43iT - 19T^{2} \) |
| 23 | \( 1 + 7.48T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6.45iT - 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 16.9iT - 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.131640071602497091103831240820, −8.389929124935318665948420299094, −7.56398215958028032427682007854, −6.39244427174115295224595962749, −5.94551004821542665031787734099, −5.35889556968374739611710428408, −4.36142071857066732301410548703, −3.47722868332658595376939499296, −2.08933638131478247430689577902, −0.74248320143640227645291254617,
0.38419237923052095631232399142, 2.20291327126239517998013240128, 3.11190158118251482157559913800, 4.08050369856766110357542105926, 5.22758383853406000213164670130, 5.98340266123638642961062834343, 6.56060012233086072406847225102, 7.06185358504936960738234776673, 8.127256758443657258287214497032, 9.193600645856779451836037275070
