L(s) = 1 | + (1.87 + 0.697i)2-s + (1.22 + 1.22i)3-s + (3.02 + 2.61i)4-s + (−5.24 − 5.24i)5-s + (1.44 + 3.14i)6-s − 5.32·7-s + (3.85 + 7.01i)8-s + 2.99i·9-s + (−6.17 − 13.4i)10-s + (12.2 − 12.2i)11-s + (0.507 + 6.90i)12-s + (−5.73 + 5.73i)13-s + (−9.98 − 3.71i)14-s − 12.8i·15-s + (2.33 + 15.8i)16-s − 23.3·17-s + ⋯ |
L(s) = 1 | + (0.937 + 0.348i)2-s + (0.408 + 0.408i)3-s + (0.757 + 0.653i)4-s + (−1.04 − 1.04i)5-s + (0.240 + 0.524i)6-s − 0.761·7-s + (0.481 + 0.876i)8-s + 0.333i·9-s + (−0.617 − 1.34i)10-s + (1.11 − 1.11i)11-s + (0.0423 + 0.575i)12-s + (−0.441 + 0.441i)13-s + (−0.713 − 0.265i)14-s − 0.856i·15-s + (0.146 + 0.989i)16-s − 1.37·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.65617 + 0.457865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65617 + 0.457865i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.87 - 0.697i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
good | 5 | \( 1 + (5.24 + 5.24i)T + 25iT^{2} \) |
| 7 | \( 1 + 5.32T + 49T^{2} \) |
| 11 | \( 1 + (-12.2 + 12.2i)T - 121iT^{2} \) |
| 13 | \( 1 + (5.73 - 5.73i)T - 169iT^{2} \) |
| 17 | \( 1 + 23.3T + 289T^{2} \) |
| 19 | \( 1 + (-11.7 - 11.7i)T + 361iT^{2} \) |
| 23 | \( 1 - 5.80T + 529T^{2} \) |
| 29 | \( 1 + (-18.3 + 18.3i)T - 841iT^{2} \) |
| 31 | \( 1 - 16.9iT - 961T^{2} \) |
| 37 | \( 1 + (-15.3 - 15.3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 29.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-33.4 + 33.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 18.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (66.9 + 66.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (27.1 - 27.1i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-65.2 + 65.2i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (37.6 + 37.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 42.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 21.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-24.1 - 24.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 52.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 21.0T + 9.40e3T^{2} \) |
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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
−15.70566566389284270361654439212, −14.38622189110549986402688963545, −13.33093332012915598089630550911, −12.19119679289400497228183444248, −11.29062680877747774680897316354, −9.182502270890562886825474041324, −8.121062887412855279674195823742, −6.48310952935779175703933109029, −4.63841283494658027264344300370, −3.52608016002707217453223146512,
2.80832534364437408566934433913, 4.19890698947182026530310114144, 6.63535586026509442723145163868, 7.28707758726511848799234937687, 9.511334292228899367970647727642, 10.97324011609666625040783275742, 12.00491840895790898239092124506, 12.96220445385298483648862829827, 14.29012281502386743278538904176, 15.10700366629264439080578999824