| L(s) = 1 | − 2-s + 1.57·3-s + 4-s + 0.329·5-s − 1.57·6-s + 7-s − 8-s − 0.507·9-s − 0.329·10-s + 3.25·11-s + 1.57·12-s + 3.60·13-s − 14-s + 0.520·15-s + 16-s − 2.15·17-s + 0.507·18-s + 0.329·20-s + 1.57·21-s − 3.25·22-s + 3.48·23-s − 1.57·24-s − 4.89·25-s − 3.60·26-s − 5.53·27-s + 28-s + 7.27·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.911·3-s + 0.5·4-s + 0.147·5-s − 0.644·6-s + 0.377·7-s − 0.353·8-s − 0.169·9-s − 0.104·10-s + 0.980·11-s + 0.455·12-s + 0.999·13-s − 0.267·14-s + 0.134·15-s + 0.250·16-s − 0.521·17-s + 0.119·18-s + 0.0736·20-s + 0.344·21-s − 0.693·22-s + 0.727·23-s − 0.322·24-s − 0.978·25-s − 0.706·26-s − 1.06·27-s + 0.188·28-s + 1.35·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.315963677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.315963677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.57T + 3T^{2} \) |
| 5 | \( 1 - 0.329T + 5T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 - 7.27T + 29T^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 - 8.60T + 37T^{2} \) |
| 41 | \( 1 + 0.542T + 41T^{2} \) |
| 43 | \( 1 + 5.42T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 - 0.375T + 67T^{2} \) |
| 71 | \( 1 - 6.95T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 + 5.46T + 83T^{2} \) |
| 89 | \( 1 - 0.179T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.342068933642970888053635839587, −7.81925960945449438706642991586, −6.86941576337360386669854827380, −6.28672171450382603148178090448, −5.47438148774259933027076168575, −4.29994385682793108081454506915, −3.59657121438415333405438295767, −2.70335552819145512581823012860, −1.87661191131047925370073000302, −0.919999169307741399171680945965,
0.919999169307741399171680945965, 1.87661191131047925370073000302, 2.70335552819145512581823012860, 3.59657121438415333405438295767, 4.29994385682793108081454506915, 5.47438148774259933027076168575, 6.28672171450382603148178090448, 6.86941576337360386669854827380, 7.81925960945449438706642991586, 8.342068933642970888053635839587
