| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 2·11-s + 3·13-s − 14-s + 16-s + 17-s + 2·22-s + 3·23-s − 3·26-s + 28-s − 3·29-s − 31-s − 32-s − 34-s + 4·37-s − 10·41-s − 43-s − 2·44-s − 3·46-s + 10·47-s + 49-s + 3·52-s − 53-s − 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.426·22-s + 0.625·23-s − 0.588·26-s + 0.188·28-s − 0.557·29-s − 0.179·31-s − 0.176·32-s − 0.171·34-s + 0.657·37-s − 1.56·41-s − 0.152·43-s − 0.301·44-s − 0.442·46-s + 1.45·47-s + 1/7·49-s + 0.416·52-s − 0.137·53-s − 0.133·56-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
| \(L(1)\) |
\(\approx\) |
\(1.484261232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484261232\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.934678094637928647151679949471, −7.06747369308095997974190290691, −6.52570024756723649510363126125, −5.61171300767616887934115577993, −5.15154377847649382881394030281, −4.10574338933960328477538929109, −3.34429936437356909372680533789, −2.46973556288306462787255308969, −1.60786010853034356947454881639, −0.66890876518947021032886689250,
0.66890876518947021032886689250, 1.60786010853034356947454881639, 2.46973556288306462787255308969, 3.34429936437356909372680533789, 4.10574338933960328477538929109, 5.15154377847649382881394030281, 5.61171300767616887934115577993, 6.52570024756723649510363126125, 7.06747369308095997974190290691, 7.934678094637928647151679949471
