| L(s) = 1 | − 2-s − 4-s + 2·7-s + 3·8-s + 11-s + 2·13-s − 2·14-s − 16-s + 2·17-s − 6·19-s − 22-s + 4·23-s − 2·26-s − 2·28-s + 6·29-s + 4·31-s − 5·32-s − 2·34-s + 6·37-s + 6·38-s + 10·41-s − 6·43-s − 44-s − 4·46-s − 8·47-s − 3·49-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s + 0.301·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.485·17-s − 1.37·19-s − 0.213·22-s + 0.834·23-s − 0.392·26-s − 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.883·32-s − 0.342·34-s + 0.986·37-s + 0.973·38-s + 1.56·41-s − 0.914·43-s − 0.150·44-s − 0.589·46-s − 1.16·47-s − 3/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.220260125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.220260125\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.830395502241705092309165842029, −8.262071324122218940913192295214, −7.74172891477451683362272747535, −6.70658497755686783298834292355, −5.90168874517427472521043428022, −4.71023634668304735532834533195, −4.39745580764377306010196421096, −3.14633475496232283891170281823, −1.79028507461184688542461496255, −0.837229374085811072713949278462,
0.837229374085811072713949278462, 1.79028507461184688542461496255, 3.14633475496232283891170281823, 4.39745580764377306010196421096, 4.71023634668304735532834533195, 5.90168874517427472521043428022, 6.70658497755686783298834292355, 7.74172891477451683362272747535, 8.262071324122218940913192295214, 8.830395502241705092309165842029
