| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 7·13-s − 14-s + 16-s + 3·17-s − 19-s − 6·23-s + 7·26-s + 28-s − 3·29-s − 4·31-s − 32-s − 3·34-s + 2·37-s + 38-s + 6·41-s + 2·43-s + 6·46-s − 3·47-s + 49-s − 7·52-s − 9·53-s − 56-s + 3·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.94·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s − 1.25·23-s + 1.37·26-s + 0.188·28-s − 0.557·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s + 0.328·37-s + 0.162·38-s + 0.937·41-s + 0.304·43-s + 0.884·46-s − 0.437·47-s + 1/7·49-s − 0.970·52-s − 1.23·53-s − 0.133·56-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9604004079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9604004079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.70595001219513465489014532165, −7.31555033374936383081326140622, −6.43510444943407556049589530817, −5.69081096820618110313289876275, −5.00528705039043826170727666195, −4.24718465980959659712670196172, −3.29114191983228101293249452846, −2.36088017065061396683721289385, −1.79332256064475750538790422979, −0.50419447856011376546608851540,
0.50419447856011376546608851540, 1.79332256064475750538790422979, 2.36088017065061396683721289385, 3.29114191983228101293249452846, 4.24718465980959659712670196172, 5.00528705039043826170727666195, 5.69081096820618110313289876275, 6.43510444943407556049589530817, 7.31555033374936383081326140622, 7.70595001219513465489014532165
