| L(s) = 1 | + 3-s + 9-s − 3·11-s − 5·13-s − 5·19-s − 9·23-s + 27-s + 10·31-s − 3·33-s + 37-s − 5·39-s + 9·41-s + 8·43-s + 3·47-s + 3·53-s − 5·57-s − 12·59-s + 8·61-s + 8·67-s − 9·69-s + 6·71-s − 2·73-s − 8·79-s + 81-s + 6·89-s + 10·93-s − 8·97-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.904·11-s − 1.38·13-s − 1.14·19-s − 1.87·23-s + 0.192·27-s + 1.79·31-s − 0.522·33-s + 0.164·37-s − 0.800·39-s + 1.40·41-s + 1.21·43-s + 0.437·47-s + 0.412·53-s − 0.662·57-s − 1.56·59-s + 1.02·61-s + 0.977·67-s − 1.08·69-s + 0.712·71-s − 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.635·89-s + 1.03·93-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.46661422619863, −14.11635835645155, −13.73480893981727, −12.98823875750730, −12.55468204739490, −12.26188176005791, −11.59936711126180, −10.94621540211994, −10.32227592057935, −10.00475528323804, −9.570393057683810, −8.849598063210694, −8.310495525126958, −7.785972464430188, −7.521718759419625, −6.761902137655151, −6.097455466162187, −5.669157732148595, −4.740385147621342, −4.444787458121438, −3.830675643166444, −2.917035449998914, −2.308210133098445, −2.149400995433471, −0.8637504835221836, 0,
0.8637504835221836, 2.149400995433471, 2.308210133098445, 2.917035449998914, 3.830675643166444, 4.444787458121438, 4.740385147621342, 5.669157732148595, 6.097455466162187, 6.761902137655151, 7.521718759419625, 7.785972464430188, 8.310495525126958, 8.849598063210694, 9.570393057683810, 10.00475528323804, 10.32227592057935, 10.94621540211994, 11.59936711126180, 12.26188176005791, 12.55468204739490, 12.98823875750730, 13.73480893981727, 14.11635835645155, 14.46661422619863
