| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s + 11-s + 4·13-s − 4·14-s − 16-s + 4·17-s + 22-s + 4·23-s − 5·25-s + 4·26-s + 4·28-s + 2·29-s − 8·31-s + 5·32-s + 4·34-s − 6·41-s − 44-s + 4·46-s − 12·47-s + 9·49-s − 5·50-s − 4·52-s + 10·53-s + 12·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s + 0.301·11-s + 1.10·13-s − 1.06·14-s − 1/4·16-s + 0.970·17-s + 0.213·22-s + 0.834·23-s − 25-s + 0.784·26-s + 0.755·28-s + 0.371·29-s − 1.43·31-s + 0.883·32-s + 0.685·34-s − 0.937·41-s − 0.150·44-s + 0.589·46-s − 1.75·47-s + 9/7·49-s − 0.707·50-s − 0.554·52-s + 1.37·53-s + 1.60·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.660876800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.660876800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−14.80313276962203, −14.37732366368343, −13.70200502702764, −13.28075088431346, −13.05588554337162, −12.35267157906488, −12.05770205421925, −11.30933003020916, −10.73767216605605, −9.960533834409158, −9.547914794786065, −9.237492980832721, −8.430412111035658, −8.076065646360248, −7.033720315618199, −6.595570892041288, −6.108515712153598, −5.402522576222803, −5.129438811057500, −3.910416818448163, −3.725239182532779, −3.278055053911491, −2.483617801830810, −1.346000477884718, −0.4534245420940150,
0.4534245420940150, 1.346000477884718, 2.483617801830810, 3.278055053911491, 3.725239182532779, 3.910416818448163, 5.129438811057500, 5.402522576222803, 6.108515712153598, 6.595570892041288, 7.033720315618199, 8.076065646360248, 8.430412111035658, 9.237492980832721, 9.547914794786065, 9.960533834409158, 10.73767216605605, 11.30933003020916, 12.05770205421925, 12.35267157906488, 13.05588554337162, 13.28075088431346, 13.70200502702764, 14.37732366368343, 14.80313276962203
