L(s) = 1 | + 5-s − 7-s − 3·9-s + 5.65·11-s + 13-s − 3.65·17-s + 4·23-s + 25-s − 2·29-s + 1.65·31-s − 35-s + 6·37-s + 7.65·41-s − 5.65·43-s − 3·45-s + 49-s − 7.65·53-s + 5.65·55-s + 6·61-s + 3·63-s + 65-s + 4·67-s + 4·71-s + 10·73-s − 5.65·77-s − 11.3·79-s + 9·81-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s + 1.70·11-s + 0.277·13-s − 0.886·17-s + 0.834·23-s + 0.200·25-s − 0.371·29-s + 0.297·31-s − 0.169·35-s + 0.986·37-s + 1.19·41-s − 0.862·43-s − 0.447·45-s + 0.142·49-s − 1.05·53-s + 0.762·55-s + 0.768·61-s + 0.377·63-s + 0.124·65-s + 0.488·67-s + 0.474·71-s + 1.17·73-s − 0.644·77-s − 1.27·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.976530094\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.976530094\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.772533081763213916044276311636, −7.88026751192434424991568866342, −6.76648695484785799005975648057, −6.40667951690595775554135824645, −5.70014498938419276630960971057, −4.71859619917603732993817228806, −3.84361625626187544714246211293, −3.01458857276786843919460598197, −2.02502697885190316641699992039, −0.834258784063352286170228816933,
0.834258784063352286170228816933, 2.02502697885190316641699992039, 3.01458857276786843919460598197, 3.84361625626187544714246211293, 4.71859619917603732993817228806, 5.70014498938419276630960971057, 6.40667951690595775554135824645, 6.76648695484785799005975648057, 7.88026751192434424991568866342, 8.772533081763213916044276311636