| L(s) = 1 | − 3-s − 5-s + 9-s + 3·11-s − 6·13-s + 15-s + 17-s − 6·19-s + 2·23-s − 4·25-s − 27-s − 3·29-s − 7·31-s − 3·33-s − 4·37-s + 6·39-s + 6·41-s + 6·43-s − 45-s − 6·47-s − 51-s − 5·53-s − 3·55-s + 6·57-s + 59-s + 6·65-s − 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s − 1.66·13-s + 0.258·15-s + 0.242·17-s − 1.37·19-s + 0.417·23-s − 4/5·25-s − 0.192·27-s − 0.557·29-s − 1.25·31-s − 0.522·33-s − 0.657·37-s + 0.960·39-s + 0.937·41-s + 0.914·43-s − 0.149·45-s − 0.875·47-s − 0.140·51-s − 0.686·53-s − 0.404·55-s + 0.794·57-s + 0.130·59-s + 0.744·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.41826884453966, −12.74829179489070, −12.59178246270743, −12.12322398205483, −11.62269837468368, −11.19945568077047, −10.73728953749527, −10.25621682659723, −9.590123588116953, −9.337869859499004, −8.801235419534050, −8.094498565648090, −7.607754017756801, −7.153152066894284, −6.798138930867857, −6.024891116537979, −5.786717498837191, −4.888149754915387, −4.704168087926790, −3.947911169079189, −3.634786030734144, −2.774418609396972, −2.063034286305888, −1.633070451527950, −0.6010066966584816, 0,
0.6010066966584816, 1.633070451527950, 2.063034286305888, 2.774418609396972, 3.634786030734144, 3.947911169079189, 4.704168087926790, 4.888149754915387, 5.786717498837191, 6.024891116537979, 6.798138930867857, 7.153152066894284, 7.607754017756801, 8.094498565648090, 8.801235419534050, 9.337869859499004, 9.590123588116953, 10.25621682659723, 10.73728953749527, 11.19945568077047, 11.62269837468368, 12.12322398205483, 12.59178246270743, 12.74829179489070, 13.41826884453966
