| L(s) = 1 | − 4.30·3-s + 2.23i·5-s + 1.47i·7-s + 9.55·9-s + 6.04i·11-s − 5.21·13-s − 9.63i·15-s + 15.7i·17-s − 4.82i·19-s − 6.35i·21-s + (2.58 + 22.8i)23-s − 5.00·25-s − 2.37·27-s − 23.4·29-s + 20.4·31-s + ⋯ |
| L(s) = 1 | − 1.43·3-s + 0.447i·5-s + 0.210i·7-s + 1.06·9-s + 0.549i·11-s − 0.401·13-s − 0.642i·15-s + 0.923i·17-s − 0.254i·19-s − 0.302i·21-s + (0.112 + 0.993i)23-s − 0.200·25-s − 0.0880·27-s − 0.809·29-s + 0.660·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4442189056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4442189056\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-2.58 - 22.8i)T \) |
| good | 3 | \( 1 + 4.30T + 9T^{2} \) |
| 7 | \( 1 - 1.47iT - 49T^{2} \) |
| 11 | \( 1 - 6.04iT - 121T^{2} \) |
| 13 | \( 1 + 5.21T + 169T^{2} \) |
| 17 | \( 1 - 15.7iT - 289T^{2} \) |
| 19 | \( 1 + 4.82iT - 361T^{2} \) |
| 29 | \( 1 + 23.4T + 841T^{2} \) |
| 31 | \( 1 - 20.4T + 961T^{2} \) |
| 37 | \( 1 - 15.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 20.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 38.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 13.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 38.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 100. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 32.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 24.1T + 5.04e3T^{2} \) |
| 73 | \( 1 - 15.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 11.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 44.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 154. iT - 9.40e3T^{2} \) |
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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
−9.757451617751572901358307051314, −8.733194890315851769474833504361, −7.64269117201784389144115724376, −6.98598625116761553835133042655, −6.16982449949975343723667044475, −5.57003445991216863527234682118, −4.75076100239010371105988848106, −3.82684799428766450278571549664, −2.50853110959106799526242864246, −1.25902073406475322939345089143,
0.18605769196419663344829685327, 0.957496746062487071995766917168, 2.46857969034507185819181195390, 3.85773869368090460241720377909, 4.79943717244478924093320573302, 5.39045627601289561280465872472, 6.15423828028294281711217011416, 6.91809644671029955079609072275, 7.73994599941179498084915042716, 8.728265455306446486759592268428
