| L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s + 2·11-s − 6·13-s + 4·14-s + 16-s + 17-s − 20-s − 2·22-s + 25-s + 6·26-s − 4·28-s + 6·29-s − 32-s − 34-s + 4·35-s − 8·37-s + 40-s + 6·41-s + 6·43-s + 2·44-s + 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s + 0.603·11-s − 1.66·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.223·20-s − 0.426·22-s + 1/5·25-s + 1.17·26-s − 0.755·28-s + 1.11·29-s − 0.176·32-s − 0.171·34-s + 0.676·35-s − 1.31·37-s + 0.158·40-s + 0.937·41-s + 0.914·43-s + 0.301·44-s + 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6869398723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6869398723\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.437374128976191160202121184212, −8.883610310076228724358554356646, −7.79075585961504557584815401628, −7.07248460911324408971461400488, −6.52251870267944311756743329514, −5.49207315012584236022884052312, −4.27523349802411276856743533388, −3.24386876800813283298871644578, −2.38847012078832059052563246572, −0.62210569401833026805129236241,
0.62210569401833026805129236241, 2.38847012078832059052563246572, 3.24386876800813283298871644578, 4.27523349802411276856743533388, 5.49207315012584236022884052312, 6.52251870267944311756743329514, 7.07248460911324408971461400488, 7.79075585961504557584815401628, 8.883610310076228724358554356646, 9.437374128976191160202121184212
