| L(s) = 1 | − 1.41i·3-s + (1.22 − 1.87i)5-s + 2.64·7-s + 0.999·9-s − 3.46i·11-s + 2.44·13-s + (−2.64 − 1.73i)15-s − 4.89·17-s − 6.48·19-s − 3.74i·21-s + (−2 − 4.58i)25-s − 5.65i·27-s + 6·29-s − 4.89·33-s + (3.24 − 4.94i)35-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s + (0.547 − 0.836i)5-s + 0.999·7-s + 0.333·9-s − 1.04i·11-s + 0.679·13-s + (−0.683 − 0.447i)15-s − 1.18·17-s − 1.48·19-s − 0.816i·21-s + (−0.400 − 0.916i)25-s − 1.08i·27-s + 1.11·29-s − 0.852·33-s + (0.547 − 0.836i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.180303317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.180303317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.22 + 1.87i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.16iT - 37T^{2} \) |
| 41 | \( 1 - 7.48iT - 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 9.16iT - 53T^{2} \) |
| 59 | \( 1 + 6.48T + 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 + 5.29T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 7.48iT - 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.582044473330414566711653401970, −8.227675128686674305108445483953, −7.25770880804222723580495381745, −6.26445079112131760915261173756, −5.87106994250193998132561792826, −4.63828406878449470912095297895, −4.16582765179697718546813043243, −2.49846753467303134878307818916, −1.67426864014841430957072511364, −0.76290827379770833970110725382,
1.67652853073332435327468360378, 2.42728961590285913405169154703, 3.76853867222418432613162180685, 4.51411878927647037488515522757, 5.07712372252929564162839018448, 6.35993821023180357030352934696, 6.78905334103169699648615795712, 7.79514342394422815881304140672, 8.653318620850220246506276694721, 9.363690090605382659455918662835
