| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 2i·7-s + i·8-s − 9-s − 6·11-s − i·12-s + 2·14-s + 16-s + 2i·17-s + i·18-s + 19-s − 2·21-s + 6i·22-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.755i·7-s + 0.353i·8-s − 0.333·9-s − 1.80·11-s − 0.288i·12-s + 0.534·14-s + 0.250·16-s + 0.485i·17-s + 0.235i·18-s + 0.229·19-s − 0.436·21-s + 1.27i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6732328246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6732328246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−8.514363960329833016188459269758, −8.195120690690702149987255975855, −7.10384103306805227964925950604, −5.92986547221285387275910221698, −5.26721247081828854607955720388, −4.67999193834739027564263963275, −3.57599299436849554045616775686, −2.75102892443070551879861934766, −2.03729841766325524846121265926, −0.24166529714376774706309971269,
1.03826223807395454268976341650, 2.48236430737732112372111426490, 3.40033153489839189689044095054, 4.56862626690688092125252795506, 5.30420021914845587654280960194, 5.95815623256053133809195794119, 7.07434882209134449263965328751, 7.37365270402183415445398658467, 8.070432006248144779169048415832, 8.748405958471388080676542011185
