| L(s) = 1 | − 2.56·2-s − 3-s + 4.56·4-s + 2.56·6-s + 3.12·7-s − 6.56·8-s + 9-s − 5.56·11-s − 4.56·12-s + 2·13-s − 8·14-s + 7.68·16-s − 1.12·17-s − 2.56·18-s − 3.12·19-s − 3.12·21-s + 14.2·22-s − 2.43·23-s + 6.56·24-s − 5.12·26-s − 27-s + 14.2·28-s + 29-s + 4·31-s − 6.56·32-s + 5.56·33-s + 2.87·34-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.577·3-s + 2.28·4-s + 1.04·6-s + 1.18·7-s − 2.31·8-s + 0.333·9-s − 1.67·11-s − 1.31·12-s + 0.554·13-s − 2.13·14-s + 1.92·16-s − 0.272·17-s − 0.603·18-s − 0.716·19-s − 0.681·21-s + 3.03·22-s − 0.508·23-s + 1.33·24-s − 1.00·26-s − 0.192·27-s + 2.69·28-s + 0.185·29-s + 0.718·31-s − 1.15·32-s + 0.968·33-s + 0.493·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5377091376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5377091376\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 1.56T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 + 6.68T + 97T^{2} \) |
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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
−8.768828903644330276023609874055, −8.456027232849861413611735095652, −7.67028272507488677244890318114, −7.14973796147253365657969554588, −6.08519516462701222910325816287, −5.35367422485844545339458319311, −4.29007800986332889679917777341, −2.62268070207715116817454999560, −1.83965249723161771003265248928, −0.64227985210694428018150375061,
0.64227985210694428018150375061, 1.83965249723161771003265248928, 2.62268070207715116817454999560, 4.29007800986332889679917777341, 5.35367422485844545339458319311, 6.08519516462701222910325816287, 7.14973796147253365657969554588, 7.67028272507488677244890318114, 8.456027232849861413611735095652, 8.768828903644330276023609874055
