| L(s) = 1 | + 3-s + 2·7-s + 9-s + 2·11-s − 2·13-s + 4·17-s + 6·19-s + 2·21-s − 6·23-s + 27-s + 10·29-s − 8·31-s + 2·33-s + 2·37-s − 2·39-s + 2·41-s − 43-s + 2·47-s − 3·49-s + 4·51-s + 10·53-s + 6·57-s − 2·59-s + 12·61-s + 2·63-s − 12·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.970·17-s + 1.37·19-s + 0.436·21-s − 1.25·23-s + 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.348·33-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.152·43-s + 0.291·47-s − 3/7·49-s + 0.560·51-s + 1.37·53-s + 0.794·57-s − 0.260·59-s + 1.53·61-s + 0.251·63-s − 1.46·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.625152381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.625152381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.97870946349124, −12.55303497490066, −11.98680502065696, −11.72756697889017, −11.35153443986785, −10.52672243623615, −10.15571714337373, −9.750008472381025, −9.318597248803314, −8.663735506194913, −8.335523278537549, −7.785422943635671, −7.340102015216856, −7.046971257653069, −6.229575411624023, −5.728951779315097, −5.211936299141150, −4.711231829264739, −4.076381985053476, −3.668523938393761, −2.981099047185216, −2.515916459755112, −1.746676993878018, −1.288528053011115, −0.6035890519012866,
0.6035890519012866, 1.288528053011115, 1.746676993878018, 2.515916459755112, 2.981099047185216, 3.668523938393761, 4.076381985053476, 4.711231829264739, 5.211936299141150, 5.728951779315097, 6.229575411624023, 7.046971257653069, 7.340102015216856, 7.785422943635671, 8.335523278537549, 8.663735506194913, 9.318597248803314, 9.750008472381025, 10.15571714337373, 10.52672243623615, 11.35153443986785, 11.72756697889017, 11.98680502065696, 12.55303497490066, 12.97870946349124
