| L(s) = 1 | + 3-s + 2·5-s − 3·7-s + 9-s + 5·13-s + 2·15-s − 4·17-s − 3·21-s + 6·23-s − 25-s + 27-s − 4·29-s − 7·31-s − 6·35-s − 37-s + 5·39-s + 11·43-s + 2·45-s + 6·47-s + 2·49-s − 4·51-s − 2·53-s + 8·59-s + 7·61-s − 3·63-s + 10·65-s + 3·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.13·7-s + 1/3·9-s + 1.38·13-s + 0.516·15-s − 0.970·17-s − 0.654·21-s + 1.25·23-s − 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.25·31-s − 1.01·35-s − 0.164·37-s + 0.800·39-s + 1.67·43-s + 0.298·45-s + 0.875·47-s + 2/7·49-s − 0.560·51-s − 0.274·53-s + 1.04·59-s + 0.896·61-s − 0.377·63-s + 1.24·65-s + 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.820701389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.820701389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.74449870876354784919108481427, −6.95164673489033155948550747340, −6.43237161286862642424315922218, −5.82030324363514557112272372301, −5.12070091467295147000498485272, −3.91554323416005883606139753293, −3.56461270377900241165705755732, −2.57627617174510397435159984953, −1.92583343092301567004078408134, −0.793650043229984169399194244063,
0.793650043229984169399194244063, 1.92583343092301567004078408134, 2.57627617174510397435159984953, 3.56461270377900241165705755732, 3.91554323416005883606139753293, 5.12070091467295147000498485272, 5.82030324363514557112272372301, 6.43237161286862642424315922218, 6.95164673489033155948550747340, 7.74449870876354784919108481427
