| L(s) = 1 | + 3-s − 5-s + 9-s + 11-s − 15-s + 17-s + 5·19-s + 25-s + 27-s − 3·29-s + 4·31-s + 33-s − 4·37-s − 7·41-s + 43-s − 45-s − 7·49-s + 51-s − 55-s + 5·57-s − 12·59-s + 6·61-s − 5·67-s − 8·71-s + 16·73-s + 75-s − 2·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.258·15-s + 0.242·17-s + 1.14·19-s + 1/5·25-s + 0.192·27-s − 0.557·29-s + 0.718·31-s + 0.174·33-s − 0.657·37-s − 1.09·41-s + 0.152·43-s − 0.149·45-s − 49-s + 0.140·51-s − 0.134·55-s + 0.662·57-s − 1.56·59-s + 0.768·61-s − 0.610·67-s − 0.949·71-s + 1.87·73-s + 0.115·75-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.88943313594129, −12.17919629461428, −11.94483093500805, −11.55467844576953, −10.95679191956531, −10.52849623594194, −9.923674950056381, −9.671748548331313, −9.014347289087659, −8.796098611764747, −8.113130455589980, −7.732407309074510, −7.436844384001321, −6.755836982410377, −6.428891452417259, −5.790046898868392, −5.056477685938824, −4.909803206705351, −4.111473375793500, −3.657526682810942, −3.202135800398570, −2.785005646500040, −1.977268176794906, −1.468699630524219, −0.8345992640795870, 0,
0.8345992640795870, 1.468699630524219, 1.977268176794906, 2.785005646500040, 3.202135800398570, 3.657526682810942, 4.111473375793500, 4.909803206705351, 5.056477685938824, 5.790046898868392, 6.428891452417259, 6.755836982410377, 7.436844384001321, 7.732407309074510, 8.113130455589980, 8.796098611764747, 9.014347289087659, 9.671748548331313, 9.923674950056381, 10.52849623594194, 10.95679191956531, 11.55467844576953, 11.94483093500805, 12.17919629461428, 12.88943313594129
