L(s) = 1 | + 2.49·3-s + (2.19 − 0.417i)5-s − 3.04i·7-s + 3.24·9-s − 5.87·11-s − 6.00i·13-s + (5.48 − 1.04i)15-s + 0.968·17-s − 1.72·19-s − 7.61i·21-s + (3.79 − 2.93i)23-s + (4.65 − 1.83i)25-s + 0.611·27-s − 3.89·29-s + 9.01i·31-s + ⋯ |
L(s) = 1 | + 1.44·3-s + (0.982 − 0.186i)5-s − 1.15i·7-s + 1.08·9-s − 1.77·11-s − 1.66i·13-s + (1.41 − 0.269i)15-s + 0.234·17-s − 0.395·19-s − 1.66i·21-s + (0.790 − 0.612i)23-s + (0.930 − 0.366i)25-s + 0.117·27-s − 0.723·29-s + 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.019090663\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.019090663\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.19 + 0.417i)T \) |
| 23 | \( 1 + (-3.79 + 2.93i)T \) |
good | 3 | \( 1 - 2.49T + 3T^{2} \) |
| 7 | \( 1 + 3.04iT - 7T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 + 6.00iT - 13T^{2} \) |
| 17 | \( 1 - 0.968T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 29 | \( 1 + 3.89T + 29T^{2} \) |
| 31 | \( 1 - 9.01iT - 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 + 9.08iT - 43T^{2} \) |
| 47 | \( 1 - 9.73T + 47T^{2} \) |
| 53 | \( 1 - 9.18T + 53T^{2} \) |
| 59 | \( 1 + 1.17iT - 59T^{2} \) |
| 61 | \( 1 - 0.951iT - 61T^{2} \) |
| 67 | \( 1 - 9.83iT - 67T^{2} \) |
| 71 | \( 1 - 8.99iT - 71T^{2} \) |
| 73 | \( 1 - 8.53iT - 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 - 5.97iT - 83T^{2} \) |
| 89 | \( 1 + 7.22iT - 89T^{2} \) |
| 97 | \( 1 - 7.51T + 97T^{2} \) |
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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
−8.853185353500390540013039636318, −8.453811048003376631051603888590, −7.53694927898759062577263005185, −7.13856725167164341929711112896, −5.67020150508071025991389829810, −5.10622675934774911406289593711, −3.87237542509908634291207222232, −2.90049127322695493061961783915, −2.37174628244994433837338735832, −0.886081392975141145366834721051,
1.98320590992773262921010921872, 2.35390132001558371894290997682, 3.14493504179343757079454099626, 4.42209183447678658405829168813, 5.46038758666877787685675817315, 6.11188441107172247804517448196, 7.29791965689786081028793159375, 7.895370778800605081134734295040, 8.927109314650822970292928644885, 9.169360639876920588688815084917