| L(s) = 1 | + 3-s + 9-s + 2·11-s − 2·13-s + 6·19-s + 27-s + 6·29-s + 10·31-s + 2·33-s − 2·39-s − 6·41-s − 8·43-s − 12·47-s + 6·53-s + 6·57-s − 6·61-s + 4·67-s − 6·71-s + 14·73-s − 4·79-s + 81-s + 6·87-s + 6·89-s + 10·93-s + 2·97-s + 2·99-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 1.37·19-s + 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.348·33-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 0.824·53-s + 0.794·57-s − 0.768·61-s + 0.488·67-s − 0.712·71-s + 1.63·73-s − 0.450·79-s + 1/9·81-s + 0.643·87-s + 0.635·89-s + 1.03·93-s + 0.203·97-s + 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.33522925041206, −12.65757014216617, −12.06649742781195, −11.85083366246513, −11.47528916306055, −10.77769228385425, −10.05973228695530, −9.937009225483891, −9.533062392688647, −8.867989807401671, −8.384498187225311, −8.114142950005745, −7.438322144834508, −7.058452591083829, −6.399262113637991, −6.234752341963230, −5.212941423321554, −4.922508783651658, −4.490454776044328, −3.646866856768204, −3.333599306812052, −2.740757375325105, −2.215123502794903, −1.366930977313561, −1.017173350136933, 0,
1.017173350136933, 1.366930977313561, 2.215123502794903, 2.740757375325105, 3.333599306812052, 3.646866856768204, 4.490454776044328, 4.922508783651658, 5.212941423321554, 6.234752341963230, 6.399262113637991, 7.058452591083829, 7.438322144834508, 8.114142950005745, 8.384498187225311, 8.867989807401671, 9.533062392688647, 9.937009225483891, 10.05973228695530, 10.77769228385425, 11.47528916306055, 11.85083366246513, 12.06649742781195, 12.65757014216617, 13.33522925041206
