| L(s) = 1 | + 1.76·3-s + 4.26·5-s + 2.37·7-s + 0.113·9-s + 6.24·11-s + 2.70·13-s + 7.51·15-s − 4.14·17-s − 2.80·19-s + 4.19·21-s + 3.25·23-s + 13.1·25-s − 5.09·27-s − 6.91·29-s + 2.09·31-s + 11.0·33-s + 10.1·35-s − 1.19·37-s + 4.76·39-s − 2.25·41-s − 4.81·43-s + 0.482·45-s + 47-s − 1.34·49-s − 7.31·51-s + 8.16·53-s + 26.6·55-s + ⋯ |
| L(s) = 1 | + 1.01·3-s + 1.90·5-s + 0.898·7-s + 0.0377·9-s + 1.88·11-s + 0.749·13-s + 1.94·15-s − 1.00·17-s − 0.642·19-s + 0.915·21-s + 0.679·23-s + 2.63·25-s − 0.980·27-s − 1.28·29-s + 0.375·31-s + 1.91·33-s + 1.71·35-s − 0.196·37-s + 0.763·39-s − 0.351·41-s − 0.734·43-s + 0.0719·45-s + 0.145·47-s − 0.192·49-s − 1.02·51-s + 1.12·53-s + 3.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.288581713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.288581713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 - 4.26T + 5T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 + 2.80T + 19T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 + 6.91T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 + 1.19T + 37T^{2} \) |
| 41 | \( 1 + 2.25T + 41T^{2} \) |
| 43 | \( 1 + 4.81T + 43T^{2} \) |
| 53 | \( 1 - 8.16T + 53T^{2} \) |
| 59 | \( 1 + 1.89T + 59T^{2} \) |
| 61 | \( 1 + 7.83T + 61T^{2} \) |
| 67 | \( 1 - 1.97T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 8.99T + 73T^{2} \) |
| 79 | \( 1 + 7.04T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 0.560T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.515813386250561102456154168335, −7.31563948546913838490750243714, −6.52267290282508771288813593166, −6.09952472283076474697858297395, −5.26223414822982387724287953045, −4.38243606846271367906051595721, −3.57003018233461429438197514871, −2.57434014059816175761507558228, −1.78414132646729359558713860806, −1.39981521225154081800141012908,
1.39981521225154081800141012908, 1.78414132646729359558713860806, 2.57434014059816175761507558228, 3.57003018233461429438197514871, 4.38243606846271367906051595721, 5.26223414822982387724287953045, 6.09952472283076474697858297395, 6.52267290282508771288813593166, 7.31563948546913838490750243714, 8.515813386250561102456154168335
