| L(s) = 1 | − 3.02·3-s − i·5-s + (−2.17 + 1.51i)7-s + 6.12·9-s + 4.71i·11-s − 2i·13-s + 3.02i·15-s + 1.12i·17-s + 4.71·19-s + (6.56 − 4.56i)21-s − 6.41i·23-s − 25-s − 9.43·27-s + 2·29-s − 3.39·31-s + ⋯ |
| L(s) = 1 | − 1.74·3-s − 0.447i·5-s + (−0.821 + 0.570i)7-s + 2.04·9-s + 1.42i·11-s − 0.554i·13-s + 0.779i·15-s + 0.272i·17-s + 1.08·19-s + (1.43 − 0.995i)21-s − 1.33i·23-s − 0.200·25-s − 1.81·27-s + 0.371·29-s − 0.609·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3901567438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3901567438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (2.17 - 1.51i)T \) |
| good | 3 | \( 1 + 3.02T + 3T^{2} \) |
| 11 | \( 1 - 4.71iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 1.12iT - 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 + 6.41iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 1.12iT - 41T^{2} \) |
| 43 | \( 1 + 0.371iT - 43T^{2} \) |
| 47 | \( 1 - 5.08T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 3.76iT - 67T^{2} \) |
| 71 | \( 1 - 7.36iT - 71T^{2} \) |
| 73 | \( 1 - 15.3iT - 73T^{2} \) |
| 79 | \( 1 + 1.32iT - 79T^{2} \) |
| 83 | \( 1 + 3.02T + 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 1.12iT - 97T^{2} \) |
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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
−9.603370333969818802252240933890, −8.615384707663902352171055494019, −7.45303071895021553723253285389, −6.82188149479723319307580435286, −6.09035613438707315368023989570, −5.34619133990266377652109212023, −4.81608253588890423218619148251, −3.84729724808249705906702477725, −2.38807377598615793810661406788, −1.02782768039119822552053321600,
0.22512145870729395016356377901, 1.29961981988691068308088171071, 3.13088492225345527491572461518, 3.87540614197770825957108875254, 4.99616782994290863821122391344, 5.79465206003779373298601196660, 6.24629992858632078786866359561, 7.08007968523879521364835856630, 7.59031013409051442404826463842, 8.986705321533615209434557231193
