| L(s) = 1 | + 2·5-s + 11-s + 6·13-s − 2·17-s − 8·19-s − 4·23-s − 25-s − 2·29-s − 8·31-s + 6·37-s − 2·41-s − 8·43-s + 4·47-s − 2·53-s + 2·55-s + 12·59-s − 10·61-s + 12·65-s + 12·67-s − 12·71-s − 10·73-s + 8·79-s − 12·83-s − 4·85-s + 10·89-s − 16·95-s + 14·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.301·11-s + 1.66·13-s − 0.485·17-s − 1.83·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s − 1.21·43-s + 0.583·47-s − 0.274·53-s + 0.269·55-s + 1.56·59-s − 1.28·61-s + 1.48·65-s + 1.46·67-s − 1.42·71-s − 1.17·73-s + 0.900·79-s − 1.31·83-s − 0.433·85-s + 1.05·89-s − 1.64·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.266906403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266906403\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.05405676417596, −13.37171827897024, −13.09128183199299, −12.83779377940921, −11.99070920638529, −11.42054550845482, −11.05561292564894, −10.44242503311811, −10.15926759613920, −9.399145632790323, −8.951068822522229, −8.562827988235652, −8.043484366754619, −7.345020791200011, −6.514290037799807, −6.348470216533932, −5.807059885873191, −5.324573638807952, −4.384030012769435, −4.001538628445454, −3.461379327003704, −2.551566912004704, −1.826684768251478, −1.617047969932086, −0.4680454153520532,
0.4680454153520532, 1.617047969932086, 1.826684768251478, 2.551566912004704, 3.461379327003704, 4.001538628445454, 4.384030012769435, 5.324573638807952, 5.807059885873191, 6.348470216533932, 6.514290037799807, 7.345020791200011, 8.043484366754619, 8.562827988235652, 8.951068822522229, 9.399145632790323, 10.15926759613920, 10.44242503311811, 11.05561292564894, 11.42054550845482, 11.99070920638529, 12.83779377940921, 13.09128183199299, 13.37171827897024, 14.05405676417596
