| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·11-s + 5·13-s + 14-s + 16-s − 4·17-s + 19-s + 3·22-s − 8·23-s − 5·26-s − 28-s − 10·29-s − 8·31-s − 32-s + 4·34-s − 4·37-s − 38-s − 9·41-s + 43-s − 3·44-s + 8·46-s + 13·47-s + 49-s + 5·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.904·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.229·19-s + 0.639·22-s − 1.66·23-s − 0.980·26-s − 0.188·28-s − 1.85·29-s − 1.43·31-s − 0.176·32-s + 0.685·34-s − 0.657·37-s − 0.162·38-s − 1.40·41-s + 0.152·43-s − 0.452·44-s + 1.17·46-s + 1.89·47-s + 1/7·49-s + 0.693·52-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
| \(L(1)\) |
\(\approx\) |
\(0.7833168521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7833168521\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.60360239435547108060639839098, −7.26479745245458794091194044201, −6.31657181581475972311114657378, −5.81359010329718717973944390744, −5.15024311747939839631576888232, −3.84188709434693402001053731625, −3.59961368606837397357328719865, −2.31447319786018017930986609257, −1.80498866260451552452278515470, −0.45262878308051786957690064227,
0.45262878308051786957690064227, 1.80498866260451552452278515470, 2.31447319786018017930986609257, 3.59961368606837397357328719865, 3.84188709434693402001053731625, 5.15024311747939839631576888232, 5.81359010329718717973944390744, 6.31657181581475972311114657378, 7.26479745245458794091194044201, 7.60360239435547108060639839098
