| L(s) = 1 | + (1.22 + 1.22i)7-s + (−4.89 + 4.89i)13-s − i·19-s − 11·31-s + (−3.67 − 3.67i)37-s + (1.22 − 1.22i)43-s − 4i·49-s − 13·61-s + (2.44 + 2.44i)67-s + (−11.0 + 11.0i)73-s − 17i·79-s − 11.9·91-s + (13.4 + 13.4i)97-s + (−11.0 + 11.0i)103-s − 17i·109-s + ⋯ |
| L(s) = 1 | + (0.462 + 0.462i)7-s + (−1.35 + 1.35i)13-s − 0.229i·19-s − 1.97·31-s + (−0.604 − 0.604i)37-s + (0.186 − 0.186i)43-s − 0.571i·49-s − 1.66·61-s + (0.299 + 0.299i)67-s + (−1.29 + 1.29i)73-s − 1.91i·79-s − 1.25·91-s + (1.36 + 1.36i)97-s + (−1.08 + 1.08i)103-s − 1.62i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4068690223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4068690223\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-1.22 - 1.22i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (4.89 - 4.89i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 11T + 31T^{2} \) |
| 37 | \( 1 + (3.67 + 3.67i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-1.22 + 1.22i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + (-2.44 - 2.44i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (11.0 - 11.0i)T - 73iT^{2} \) |
| 79 | \( 1 + 17iT - 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-13.4 - 13.4i)T + 97iT^{2} \) |
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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
−9.132495965148806275896837623065, −8.629856540489657938499963419881, −7.45086169743203429007393729770, −7.15981511483206436744579642454, −6.10787571438957075802340105980, −5.22542882725966761758957259755, −4.59809161148752593580927683577, −3.64864297967829872559789480915, −2.41696602659429699646539129175, −1.72865510021044376903729497515,
0.12303116172073609757013029516, 1.54021247119545813487700810770, 2.69796635761054163103360855204, 3.58457881358249008689314282915, 4.64754022954800003218814971574, 5.28012743065022397245234477147, 6.07869715635040277003724155422, 7.28209358013698550447797867391, 7.55371624264123097765029511813, 8.377200720828658927436386365996
