| L(s) = 1 | − 0.381·2-s − 3-s − 1.85·4-s + 0.381·6-s − 7-s + 1.47·8-s + 9-s + 11-s + 1.85·12-s − 1.38·13-s + 0.381·14-s + 3.14·16-s + 3.23·17-s − 0.381·18-s − 3.61·19-s + 21-s − 0.381·22-s − 5.47·23-s − 1.47·24-s + 0.527·26-s − 27-s + 1.85·28-s + 3.47·31-s − 4.14·32-s − 33-s − 1.23·34-s − 1.85·36-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.577·3-s − 0.927·4-s + 0.155·6-s − 0.377·7-s + 0.520·8-s + 0.333·9-s + 0.301·11-s + 0.535·12-s − 0.383·13-s + 0.102·14-s + 0.786·16-s + 0.784·17-s − 0.0900·18-s − 0.830·19-s + 0.218·21-s − 0.0814·22-s − 1.14·23-s − 0.300·24-s + 0.103·26-s − 0.192·27-s + 0.350·28-s + 0.623·31-s − 0.732·32-s − 0.174·33-s − 0.211·34-s − 0.309·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 3.47T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 - 1.76T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 + 2.52T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 - 8.70T + 79T^{2} \) |
| 83 | \( 1 + 9.32T + 83T^{2} \) |
| 89 | \( 1 - 8.52T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−7.81695846909206984793347855220, −7.10942579367595004377057786859, −6.21078304380745568505946215569, −5.65447531471983900040620298249, −4.83042682466352584380132828085, −4.14507750184375701783370421954, −3.46431143452557850385865491660, −2.19652887720665176372118496703, −1.02558883215710584666038086395, 0,
1.02558883215710584666038086395, 2.19652887720665176372118496703, 3.46431143452557850385865491660, 4.14507750184375701783370421954, 4.83042682466352584380132828085, 5.65447531471983900040620298249, 6.21078304380745568505946215569, 7.10942579367595004377057786859, 7.81695846909206984793347855220
