L(s) = 1 | + 2-s − i·3-s + 4-s + (−1.28 − 1.82i)5-s − i·6-s + 1.25i·7-s + 8-s − 9-s + (−1.28 − 1.82i)10-s − 5.46·11-s − i·12-s − 4.70·13-s + 1.25i·14-s + (−1.82 + 1.28i)15-s + 16-s − 1.19·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.5·4-s + (−0.576 − 0.817i)5-s − 0.408i·6-s + 0.474i·7-s + 0.353·8-s − 0.333·9-s + (−0.407 − 0.577i)10-s − 1.64·11-s − 0.288i·12-s − 1.30·13-s + 0.335i·14-s + (−0.471 + 0.332i)15-s + 0.250·16-s − 0.289·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6194148751\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6194148751\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.28 + 1.82i)T \) |
| 37 | \( 1 + (-4.58 - 3.99i)T \) |
good | 7 | \( 1 - 1.25iT - 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 + 1.22iT - 19T^{2} \) |
| 23 | \( 1 + 1.91T + 23T^{2} \) |
| 29 | \( 1 + 5.08iT - 29T^{2} \) |
| 31 | \( 1 + 3.43iT - 31T^{2} \) |
| 41 | \( 1 + 4.46T + 41T^{2} \) |
| 43 | \( 1 + 1.34T + 43T^{2} \) |
| 47 | \( 1 + 1.90iT - 47T^{2} \) |
| 53 | \( 1 - 1.11iT - 53T^{2} \) |
| 59 | \( 1 + 8.99iT - 59T^{2} \) |
| 61 | \( 1 + 6.21iT - 61T^{2} \) |
| 67 | \( 1 + 4.21iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 10.2iT - 73T^{2} \) |
| 79 | \( 1 - 1.00iT - 79T^{2} \) |
| 83 | \( 1 - 0.754iT - 83T^{2} \) |
| 89 | \( 1 + 11.4iT - 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−9.422045354358090439607578307350, −8.226229942455236248795994389228, −7.83017916652140995952230429575, −6.96883303126031262791197836405, −5.77166580577481919138936719478, −5.10607013772632561752212570084, −4.35841781887412390122488902659, −2.91338057698168235489672851730, −2.10344248839087253163159067300, −0.19219244230822939521146451841,
2.42357080525057391300289368080, 3.14426536519556374543904364270, 4.21355866146750344046344004900, 4.97543384226475295514038212322, 5.85508498874380110084844083059, 7.10815184876312015545905880077, 7.51503447930169889308262889851, 8.453073043672025298812660352389, 9.807606705730415070276713904940, 10.49319815558834193198073595292