L(s) = 1 | + 2-s − 2.44·3-s + 4-s + 3.56·5-s − 2.44·6-s + 3.03·7-s + 8-s + 2.99·9-s + 3.56·10-s + 11-s − 2.44·12-s − 2.39·13-s + 3.03·14-s − 8.71·15-s + 16-s + 2.80·17-s + 2.99·18-s − 3.07·19-s + 3.56·20-s − 7.42·21-s + 22-s + 0.184·23-s − 2.44·24-s + 7.68·25-s − 2.39·26-s + 0.0188·27-s + 3.03·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s + 1.59·5-s − 0.999·6-s + 1.14·7-s + 0.353·8-s + 0.997·9-s + 1.12·10-s + 0.301·11-s − 0.706·12-s − 0.664·13-s + 0.810·14-s − 2.25·15-s + 0.250·16-s + 0.681·17-s + 0.705·18-s − 0.705·19-s + 0.796·20-s − 1.62·21-s + 0.213·22-s + 0.0384·23-s − 0.499·24-s + 1.53·25-s − 0.469·26-s + 0.00363·27-s + 0.573·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.111722834\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.111722834\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 - 3.03T + 7T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 0.184T + 23T^{2} \) |
| 29 | \( 1 - 7.74T + 29T^{2} \) |
| 31 | \( 1 - 0.760T + 31T^{2} \) |
| 37 | \( 1 + 2.03T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 6.70T + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 + 1.18T + 67T^{2} \) |
| 71 | \( 1 - 4.06T + 71T^{2} \) |
| 73 | \( 1 - 8.89T + 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 - 8.57T + 83T^{2} \) |
| 89 | \( 1 - 4.06T + 89T^{2} \) |
| 97 | \( 1 + 1.65T + 97T^{2} \) |
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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
−8.302271562370261967075825983857, −7.33817646387068010612037854543, −6.48813108859936876563167952662, −6.10358426393714239998805215417, −5.26956661689554373909106977175, −5.02566566373710055095294317699, −4.22844285266917619174478428048, −2.75312741889965682851482241195, −1.87310970210375950801068035890, −1.03560751338105346054642001370,
1.03560751338105346054642001370, 1.87310970210375950801068035890, 2.75312741889965682851482241195, 4.22844285266917619174478428048, 5.02566566373710055095294317699, 5.26956661689554373909106977175, 6.10358426393714239998805215417, 6.48813108859936876563167952662, 7.33817646387068010612037854543, 8.302271562370261967075825983857