L(s) = 1 | − i·2-s + 2.24·3-s − 4-s + i·5-s − 2.24i·6-s + 1.52·7-s + i·8-s + 2.05·9-s + 10-s + 2.71·11-s − 2.24·12-s + 0.941i·13-s − 1.52i·14-s + 2.24i·15-s + 16-s − 4.83i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.29·3-s − 0.5·4-s + 0.447i·5-s − 0.918i·6-s + 0.578·7-s + 0.353i·8-s + 0.686·9-s + 0.316·10-s + 0.820·11-s − 0.649·12-s + 0.261i·13-s − 0.408i·14-s + 0.580i·15-s + 0.250·16-s − 1.17i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.90194 - 0.675572i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.90194 - 0.675572i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 + (4.71 + 3.83i)T \) |
good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 - 0.941iT - 13T^{2} \) |
| 17 | \( 1 + 4.83iT - 17T^{2} \) |
| 19 | \( 1 + 0.249iT - 19T^{2} \) |
| 23 | \( 1 - 0.941iT - 23T^{2} \) |
| 29 | \( 1 - 0.719iT - 29T^{2} \) |
| 31 | \( 1 - 4.02iT - 31T^{2} \) |
| 41 | \( 1 + 8.27T + 41T^{2} \) |
| 43 | \( 1 - 2.71iT - 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 - 8.39T + 53T^{2} \) |
| 59 | \( 1 + 7.30iT - 59T^{2} \) |
| 61 | \( 1 - 6.83iT - 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 + 3.05T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 + 1.75iT - 79T^{2} \) |
| 83 | \( 1 - 0.131T + 83T^{2} \) |
| 89 | \( 1 + 8.99iT - 89T^{2} \) |
| 97 | \( 1 - 16.2iT - 97T^{2} \) |
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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
−11.40097255173977897218454835291, −10.30541777662529068239911962754, −9.339294503359891899860335695549, −8.760698868357912906111203207788, −7.77576517692004481346599779668, −6.77737689717194813490525593769, −5.09980658806681565630354670580, −3.84498652546583190383149282653, −2.92521037072475036209692270690, −1.75099906028526191248994240465,
1.75434877437549426617228037769, 3.46476932458775479750475337363, 4.43998740984593056475423863170, 5.73125675866419033039564940374, 6.93214665642514648815940124060, 8.108655394598742955959805405811, 8.454535408731731399372285061703, 9.305243594999972273512421965601, 10.26269405315130133490366140973, 11.60601201637352154737814129942