L(s) = 1 | − 1.17i·3-s − 2.23i·5-s + 1.62·9-s + 6.62·11-s − 5.64i·13-s − 2.62·15-s + 7.99i·17-s − 5.00·25-s − 5.42i·27-s + 0.623·29-s − 7.77i·33-s − 6.62·39-s − 3.63i·45-s − 12.4i·47-s + 9.37·51-s + ⋯ |
L(s) = 1 | − 0.677i·3-s − 0.999i·5-s + 0.541·9-s + 1.99·11-s − 1.56i·13-s − 0.677·15-s + 1.93i·17-s − 1.00·25-s − 1.04i·27-s + 0.115·29-s − 1.35i·33-s − 1.06·39-s − 0.541i·45-s − 1.81i·47-s + 1.31·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29985 - 1.29985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29985 - 1.29985i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.17iT - 3T^{2} \) |
| 11 | \( 1 - 6.62T + 11T^{2} \) |
| 13 | \( 1 + 5.64iT - 13T^{2} \) |
| 17 | \( 1 - 7.99iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 0.623T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12.4iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 12.6iT - 97T^{2} \) |
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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
−9.797400572354215322609061772179, −8.724248098356151522652424591881, −8.274345815145422732304616723615, −7.28624704887873087093332972635, −6.33451030073734265750581141732, −5.64489316615954586718782898308, −4.34207175843872856157003739856, −3.63267601031804595877084156077, −1.78369571951982602566572462914, −0.995492384960664786365509229933,
1.59282582991826230632506242780, 3.03018330024072510181298466187, 4.07550942877167258480113261564, 4.62364155774651740206829018616, 6.13161152280342265318704519077, 6.90330749413495969910713994286, 7.33979240469291008243626253616, 8.989329318705106512342297596506, 9.402930681903663592093809750893, 10.01130762203300648473145341076