| L(s) = 1 | + (2.40 + 1.79i)3-s + 10.2·7-s + (2.53 + 8.63i)9-s + 8.19i·11-s + 13.5·13-s − 15.4i·17-s − 25.4·19-s + (24.5 + 18.3i)21-s + 17.9i·23-s + (−9.44 + 25.2i)27-s − 42.0i·29-s + 38.4·31-s + (−14.7 + 19.6i)33-s − 11.8·37-s + (32.6 + 24.4i)39-s + ⋯ |
| L(s) = 1 | + (0.800 + 0.599i)3-s + 1.45·7-s + (0.281 + 0.959i)9-s + 0.744i·11-s + 1.04·13-s − 0.910i·17-s − 1.34·19-s + (1.16 + 0.874i)21-s + 0.778i·23-s + (−0.349 + 0.936i)27-s − 1.44i·29-s + 1.24·31-s + (−0.446 + 0.596i)33-s − 0.319·37-s + (0.836 + 0.626i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.860833077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.860833077\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.40 - 1.79i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 10.2T + 49T^{2} \) |
| 11 | \( 1 - 8.19iT - 121T^{2} \) |
| 13 | \( 1 - 13.5T + 169T^{2} \) |
| 17 | \( 1 + 15.4iT - 289T^{2} \) |
| 19 | \( 1 + 25.4T + 361T^{2} \) |
| 23 | \( 1 - 17.9iT - 529T^{2} \) |
| 29 | \( 1 + 42.0iT - 841T^{2} \) |
| 31 | \( 1 - 38.4T + 961T^{2} \) |
| 37 | \( 1 + 11.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 46.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 54.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 43.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 82.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 45.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 93.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 34.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 68.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 44.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 11.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 144. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 63.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 63.9T + 9.40e3T^{2} \) |
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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
−10.58953563504455430057262401094, −9.665016260298097542476431633997, −8.751245106138586341009418177976, −8.086070023239151902915223135329, −7.34330519793743827427034097569, −5.91676484710645086156049966187, −4.65025192849903249621700162934, −4.20926108443920388008534871197, −2.67548756708981333907456068506, −1.57006860477106477789126392670,
1.14294089158604044627531988524, 2.16637102857668200310979596887, 3.55981847131357658342185960232, 4.55952775177785512324341643841, 5.94100625745403223253278803048, 6.76477797341254474777670813198, 8.077676853451544277887997244843, 8.364485003669870816213266360013, 9.054941019527095809242554049703, 10.65109843223178169237644541774
