| L(s) = 1 | − 2·3-s − 5-s + 9-s − 4·11-s + 13-s + 2·15-s − 4·17-s − 6·19-s + 6·23-s + 25-s + 4·27-s + 6·29-s − 4·31-s + 8·33-s − 2·37-s − 2·39-s − 45-s − 8·47-s + 8·51-s − 12·53-s + 4·55-s + 12·57-s + 6·59-s + 2·61-s − 65-s + 12·67-s − 12·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.516·15-s − 0.970·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s + 1.39·33-s − 0.328·37-s − 0.320·39-s − 0.149·45-s − 1.16·47-s + 1.12·51-s − 1.64·53-s + 0.539·55-s + 1.58·57-s + 0.781·59-s + 0.256·61-s − 0.124·65-s + 1.46·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−15.76718081470887, −15.26553409255108, −14.60489666475991, −14.04624114904146, −13.18366503602733, −12.77756849491394, −12.57912002949984, −11.65172865070685, −11.16926696233821, −10.93501665740096, −10.39227335088611, −9.798494300384872, −8.818318994260943, −8.499498126544043, −7.894214855168057, −7.015895136279723, −6.642712475642152, −6.071638867533068, −5.311435779843356, −4.833263725378895, −4.382492638967177, −3.396048197568522, −2.697672664335458, −1.871747894377714, −0.7074914176720955, 0,
0.7074914176720955, 1.871747894377714, 2.697672664335458, 3.396048197568522, 4.382492638967177, 4.833263725378895, 5.311435779843356, 6.071638867533068, 6.642712475642152, 7.015895136279723, 7.894214855168057, 8.499498126544043, 8.818318994260943, 9.798494300384872, 10.39227335088611, 10.93501665740096, 11.16926696233821, 11.65172865070685, 12.57912002949984, 12.77756849491394, 13.18366503602733, 14.04624114904146, 14.60489666475991, 15.26553409255108, 15.76718081470887
