| L(s) = 1 | − 2.23i·5-s − 4.23i·7-s − 3.42i·11-s − 13.7·13-s + 5.48i·17-s + 4.66i·19-s + (−15.0 − 17.3i)23-s − 5.00·25-s − 51.3·29-s + 36.0·31-s − 9.46·35-s + 70.1i·37-s + 29.1·41-s + 24.5i·43-s − 58.9·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 0.604i·7-s − 0.310i·11-s − 1.05·13-s + 0.322i·17-s + 0.245i·19-s + (−0.655 − 0.755i)23-s − 0.200·25-s − 1.77·29-s + 1.16·31-s − 0.270·35-s + 1.89i·37-s + 0.709·41-s + 0.571i·43-s − 1.25·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.230493621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230493621\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (15.0 + 17.3i)T \) |
| good | 7 | \( 1 + 4.23iT - 49T^{2} \) |
| 11 | \( 1 + 3.42iT - 121T^{2} \) |
| 13 | \( 1 + 13.7T + 169T^{2} \) |
| 17 | \( 1 - 5.48iT - 289T^{2} \) |
| 19 | \( 1 - 4.66iT - 361T^{2} \) |
| 29 | \( 1 + 51.3T + 841T^{2} \) |
| 31 | \( 1 - 36.0T + 961T^{2} \) |
| 37 | \( 1 - 70.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 29.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 24.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 58.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 26.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 8.79T + 3.48e3T^{2} \) |
| 61 | \( 1 + 30.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 5.12iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 45.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 75.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 62.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 137. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 64.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 171. iT - 9.40e3T^{2} \) |
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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
−8.087118122950082789457838596124, −7.84357191732574736995597546271, −6.82603553693859036071303747058, −6.19745828213646499521918580174, −5.26655342159219878920673715174, −4.56246364064651643609058971502, −3.85383850817313380566462519858, −2.84881423594704476204763203422, −1.83639480582876473579714610707, −0.75134131099948034715388911618,
0.32749493661137341539006497202, 1.93009297606483553594072050214, 2.49882463280694044674451662518, 3.51420185436684578073491941169, 4.37056925203095588430455414264, 5.33603758839043357012304579791, 5.81931909914132764605861258323, 6.84091244849231442784681331511, 7.41164142710108095135800857298, 8.024854930965542399547070939306
