| L(s) = 1 | − 3-s + 9-s + 11-s − 2·13-s − 2·17-s + 6·19-s − 27-s − 6·29-s − 4·31-s − 33-s + 2·39-s − 10·41-s − 8·43-s + 8·47-s + 2·51-s − 14·53-s − 6·57-s − 2·59-s + 14·67-s + 4·71-s − 2·73-s + 81-s − 12·83-s + 6·87-s + 4·93-s − 2·97-s + 99-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.485·17-s + 1.37·19-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s + 0.320·39-s − 1.56·41-s − 1.21·43-s + 1.16·47-s + 0.280·51-s − 1.92·53-s − 0.794·57-s − 0.260·59-s + 1.71·67-s + 0.474·71-s − 0.234·73-s + 1/9·81-s − 1.31·83-s + 0.643·87-s + 0.414·93-s − 0.203·97-s + 0.100·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.92938146888095, −12.27603824711647, −11.89722940946751, −11.52409417944101, −11.05577043582722, −10.72436517974140, −9.985420157494646, −9.698591659400442, −9.356380693252676, −8.725211772267793, −8.257100344707117, −7.634466032153455, −7.215964243470766, −6.875099000275486, −6.308327615033953, −5.750232329484058, −5.283396323388192, −4.906303308039398, −4.394269092100787, −3.605013310892342, −3.393732135358144, −2.623752704440763, −1.860872614878326, −1.519832422667589, −0.6509494155770529, 0,
0.6509494155770529, 1.519832422667589, 1.860872614878326, 2.623752704440763, 3.393732135358144, 3.605013310892342, 4.394269092100787, 4.906303308039398, 5.283396323388192, 5.750232329484058, 6.308327615033953, 6.875099000275486, 7.215964243470766, 7.634466032153455, 8.257100344707117, 8.725211772267793, 9.356380693252676, 9.698591659400442, 9.985420157494646, 10.72436517974140, 11.05577043582722, 11.52409417944101, 11.89722940946751, 12.27603824711647, 12.92938146888095
