| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s − 11-s + 13-s + 4·14-s + 16-s − 7·17-s − 3·19-s − 22-s + 26-s + 4·28-s + 4·29-s + 6·31-s + 32-s − 7·34-s + 8·37-s − 3·38-s + 5·41-s + 4·43-s − 44-s + 12·47-s + 9·49-s + 52-s − 10·53-s + 4·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 0.301·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 1.69·17-s − 0.688·19-s − 0.213·22-s + 0.196·26-s + 0.755·28-s + 0.742·29-s + 1.07·31-s + 0.176·32-s − 1.20·34-s + 1.31·37-s − 0.486·38-s + 0.780·41-s + 0.609·43-s − 0.150·44-s + 1.75·47-s + 9/7·49-s + 0.138·52-s − 1.37·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.850139936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.850139936\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−8.068440123640182722200400877380, −7.41691486979270773030871178387, −6.50521431717083753615397117222, −5.98746904972474655629584859885, −4.96760721758458728586149471633, −4.53759820208697751196598543860, −3.96702197823077873661896373807, −2.60093734673257549191785370445, −2.13696100737455622049237903573, −0.962595304245422244640426107980,
0.962595304245422244640426107980, 2.13696100737455622049237903573, 2.60093734673257549191785370445, 3.96702197823077873661896373807, 4.53759820208697751196598543860, 4.96760721758458728586149471633, 5.98746904972474655629584859885, 6.50521431717083753615397117222, 7.41691486979270773030871178387, 8.068440123640182722200400877380
