L(s) = 1 | − 1.30i·2-s − 2.30i·3-s + 0.302·4-s − 3·6-s + 0.697i·7-s − 3i·8-s − 2.30·9-s − 11-s − 0.697i·12-s − 5i·13-s + 0.908·14-s − 3.30·16-s + 6.90i·17-s + 3.00i·18-s + 19-s + ⋯ |
L(s) = 1 | − 0.921i·2-s − 1.32i·3-s + 0.151·4-s − 1.22·6-s + 0.263i·7-s − 1.06i·8-s − 0.767·9-s − 0.301·11-s − 0.201i·12-s − 1.38i·13-s + 0.242·14-s − 0.825·16-s + 1.67i·17-s + 0.707i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.320310 - 1.35685i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.320310 - 1.35685i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 1.30iT - 2T^{2} \) |
| 3 | \( 1 + 2.30iT - 3T^{2} \) |
| 7 | \( 1 - 0.697iT - 7T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 6.90iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 7.30iT - 23T^{2} \) |
| 29 | \( 1 + 0.908T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.39iT - 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 + 7.21iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 1.30iT - 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 - 7.90iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 3.51iT - 83T^{2} \) |
| 89 | \( 1 + 1.69T + 89T^{2} \) |
| 97 | \( 1 - 15.3iT - 97T^{2} \) |
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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
−11.79420417443743460785915888296, −10.63460384639914048019816236784, −9.961965054633130232765915284440, −8.395217590334114805737362695225, −7.62612870580235544207123958487, −6.57474469262712668741942296990, −5.59839985057147909381453898575, −3.59415888423743951655138386265, −2.38064580010521595927257501504, −1.18188204406964386823878367138,
2.70833761404582303362732351325, 4.39173213823727211477522426582, 5.04780809667250435484143079337, 6.41503270745777903545570645784, 7.26323469200166635535524155741, 8.503618091816844513664367368989, 9.399738631690419503626370767692, 10.30177341661500974213568475667, 11.25971529994801652939818165244, 11.97490804885884145918597945887