| L(s) = 1 | + i·2-s − 3-s − 4-s − i·6-s + 4.56i·7-s − i·8-s + 9-s − 2.56i·11-s + 12-s + (−0.561 + 3.56i)13-s − 4.56·14-s + 16-s + 5.68·17-s + i·18-s + 4.68i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.408i·6-s + 1.72i·7-s − 0.353i·8-s + 0.333·9-s − 0.772i·11-s + 0.288·12-s + (−0.155 + 0.987i)13-s − 1.21·14-s + 0.250·16-s + 1.37·17-s + 0.235i·18-s + 1.07i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.103760164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.103760164\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (0.561 - 3.56i)T \) |
| good | 7 | \( 1 - 4.56iT - 7T^{2} \) |
| 11 | \( 1 + 2.56iT - 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 - 4.68iT - 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 1.43iT - 31T^{2} \) |
| 37 | \( 1 - 0.438iT - 37T^{2} \) |
| 41 | \( 1 + 0.438iT - 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 + 0.123iT - 47T^{2} \) |
| 53 | \( 1 + 5T + 53T^{2} \) |
| 59 | \( 1 + 5.43iT - 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 4.12iT - 67T^{2} \) |
| 71 | \( 1 - 14.9iT - 71T^{2} \) |
| 73 | \( 1 - 1.12iT - 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 13.9iT - 83T^{2} \) |
| 89 | \( 1 - 3.12iT - 89T^{2} \) |
| 97 | \( 1 + 4.87iT - 97T^{2} \) |
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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
−9.453898158740055210142693075445, −8.641189041201898621176592450475, −8.121051194842610961237989955696, −7.04107517363033604328719719541, −6.26825386486825225875522654831, −5.52379825279034648862053126220, −5.20262093991857535325355008424, −3.89533824064078751141290343394, −2.82602794867711892696993896651, −1.42796548652683955988655969243,
0.49252037998692786679075801993, 1.36461604062087693313864977852, 2.92838558265745398829551065403, 3.79041230182617529228337775763, 4.70983829431609049677392815681, 5.29002437628318905451648968566, 6.52873488233185045004801022645, 7.40871082768283801511298022597, 7.77008746285127894876885693686, 9.055058705562496729376947953835
