| L(s) = 1 | − 3-s − 7-s + 9-s − 6·11-s + 2·13-s − 6·17-s − 2·19-s + 21-s − 23-s − 5·25-s − 27-s − 6·29-s − 8·31-s + 6·33-s + 8·37-s − 2·39-s + 6·41-s − 2·43-s + 49-s + 6·51-s − 12·53-s + 2·57-s + 8·61-s − 63-s + 10·67-s + 69-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 1.45·17-s − 0.458·19-s + 0.218·21-s − 0.208·23-s − 25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.04·33-s + 1.31·37-s − 0.320·39-s + 0.937·41-s − 0.304·43-s + 1/7·49-s + 0.840·51-s − 1.64·53-s + 0.264·57-s + 1.02·61-s − 0.125·63-s + 1.22·67-s + 0.120·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5236844208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5236844208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.82649995078818070298342819650, −7.18099316943188202754165388843, −6.36272639486136673232771749791, −5.78141933141309088094107707625, −5.18194655883271019147819414738, −4.33522559556098348826680369969, −3.64256650828408249010696952250, −2.53985125358816544530966024865, −1.90540760796015316102336228335, −0.34899484347838176474436710685,
0.34899484347838176474436710685, 1.90540760796015316102336228335, 2.53985125358816544530966024865, 3.64256650828408249010696952250, 4.33522559556098348826680369969, 5.18194655883271019147819414738, 5.78141933141309088094107707625, 6.36272639486136673232771749791, 7.18099316943188202754165388843, 7.82649995078818070298342819650
