| L(s) = 1 | + 2·3-s + 4·5-s + 7-s + 9-s − 4·11-s + 2·13-s + 8·15-s + 4·17-s + 2·21-s + 11·25-s − 4·27-s − 2·29-s + 6·31-s − 8·33-s + 4·35-s + 2·37-s + 4·39-s − 6·41-s + 4·43-s + 4·45-s + 10·47-s + 49-s + 8·51-s + 6·53-s − 16·55-s − 2·59-s + 8·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 2.06·15-s + 0.970·17-s + 0.436·21-s + 11/5·25-s − 0.769·27-s − 0.371·29-s + 1.07·31-s − 1.39·33-s + 0.676·35-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.596·45-s + 1.45·47-s + 1/7·49-s + 1.12·51-s + 0.824·53-s − 2.15·55-s − 0.260·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.030290936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.030290936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.02032873396729, −12.76105231906669, −12.05706031361270, −11.49266786408184, −10.78738978563893, −10.51131363992201, −9.979783064859620, −9.650331484318714, −9.226508841080519, −8.587322381932145, −8.365995019376686, −7.835002580847693, −7.324860163121497, −6.736383845738011, −6.058946670033123, −5.702217639355876, −5.256065660003314, −4.838896883042122, −3.942447856403100, −3.461317660044979, −2.702720475962380, −2.490301486839100, −2.011802199372651, −1.318579733528392, −0.7157079764298335,
0.7157079764298335, 1.318579733528392, 2.011802199372651, 2.490301486839100, 2.702720475962380, 3.461317660044979, 3.942447856403100, 4.838896883042122, 5.256065660003314, 5.702217639355876, 6.058946670033123, 6.736383845738011, 7.324860163121497, 7.835002580847693, 8.365995019376686, 8.587322381932145, 9.226508841080519, 9.650331484318714, 9.979783064859620, 10.51131363992201, 10.78738978563893, 11.49266786408184, 12.05706031361270, 12.76105231906669, 13.02032873396729
