| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s − 11-s − 2·13-s − 2·15-s − 4·17-s + 6·19-s + 2·21-s − 25-s − 27-s + 8·31-s + 33-s − 4·35-s − 10·37-s + 2·39-s − 8·41-s + 2·43-s + 2·45-s + 8·47-s − 3·49-s + 4·51-s − 2·53-s − 2·55-s − 6·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.970·17-s + 1.37·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s + 1.43·31-s + 0.174·33-s − 0.676·35-s − 1.64·37-s + 0.320·39-s − 1.24·41-s + 0.304·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s − 0.269·55-s − 0.794·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.479803823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479803823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.56109034095594, −13.29517151723232, −12.69213393911131, −12.02346730295794, −11.93325035714968, −11.21200459568841, −10.60272393316834, −10.11373484222034, −9.899048692944655, −9.290112409962489, −8.914992816386997, −8.171577575189203, −7.577495998518472, −6.978842254908827, −6.557143885778139, −6.153353960847992, −5.479073536625487, −5.078239930767316, −4.646073384599693, −3.713230099858039, −3.275077229230644, −2.441189916012835, −2.061460683420310, −1.172425415975873, −0.4096804924984696,
0.4096804924984696, 1.172425415975873, 2.061460683420310, 2.441189916012835, 3.275077229230644, 3.713230099858039, 4.646073384599693, 5.078239930767316, 5.479073536625487, 6.153353960847992, 6.557143885778139, 6.978842254908827, 7.577495998518472, 8.171577575189203, 8.914992816386997, 9.290112409962489, 9.899048692944655, 10.11373484222034, 10.60272393316834, 11.21200459568841, 11.93325035714968, 12.02346730295794, 12.69213393911131, 13.29517151723232, 13.56109034095594
