| L(s) = 1 | − 1.73i·3-s − 3.08·5-s − 2.99·9-s + 12.2i·11-s + 5.33i·15-s − 15.0·17-s − 24.5i·19-s + 8.51i·23-s − 15.4·25-s + 5.19i·27-s − 24.4·29-s − 3.75i·31-s + 21.2·33-s + 52.4·37-s + 39.0·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.616·5-s − 0.333·9-s + 1.11i·11-s + 0.355i·15-s − 0.887·17-s − 1.29i·19-s + 0.370i·23-s − 0.619·25-s + 0.192i·27-s − 0.844·29-s − 0.120i·31-s + 0.643·33-s + 1.41·37-s + 0.953·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.347131855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.347131855\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.08T + 25T^{2} \) |
| 11 | \( 1 - 12.2iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 15.0T + 289T^{2} \) |
| 19 | \( 1 + 24.5iT - 361T^{2} \) |
| 23 | \( 1 - 8.51iT - 529T^{2} \) |
| 29 | \( 1 + 24.4T + 841T^{2} \) |
| 31 | \( 1 + 3.75iT - 961T^{2} \) |
| 37 | \( 1 - 52.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 39.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 27.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 21.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 11.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 42.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 29.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 73.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 42.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 23.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 83.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 148.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 186.T + 9.40e3T^{2} \) |
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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
−8.779058791159775342830992792950, −7.68597338297061724588967674419, −7.36702808066276488186373318310, −6.58250785053731909945468840693, −5.66616781807950182003719074407, −4.62032153509005319565462432694, −4.01196415197555433480959668295, −2.73042008745469381300123129485, −1.92730300080200392340613779413, −0.55929956857743725713379986441,
0.60859400626210769332674704209, 2.14448872057675879423244490840, 3.33760702428357902226630605353, 3.93725313556305781825995639281, 4.76465153890887823826987850357, 5.83628324918865413870093828695, 6.31976577567472739503370425911, 7.57543228976301826630278993099, 8.103805728999319858838791011160, 8.861010052393703147226365006437
