L(s) = 1 | + 0.751·2-s − 1.43·4-s + 0.315·7-s − 2.58·8-s + 4.34·11-s + 4.58·13-s + 0.236·14-s + 0.933·16-s − 0.917·17-s + 2.76·19-s + 3.26·22-s + 23-s + 3.44·26-s − 0.452·28-s − 7.03·29-s − 0.867·31-s + 5.86·32-s − 0.689·34-s + 4.68·37-s + 2.07·38-s − 4.69·41-s + 9.08·43-s − 6.24·44-s + 0.751·46-s − 8.24·47-s − 6.90·49-s − 6.57·52-s + ⋯ |
L(s) = 1 | + 0.531·2-s − 0.717·4-s + 0.119·7-s − 0.912·8-s + 1.31·11-s + 1.27·13-s + 0.0632·14-s + 0.233·16-s − 0.222·17-s + 0.634·19-s + 0.696·22-s + 0.208·23-s + 0.674·26-s − 0.0854·28-s − 1.30·29-s − 0.155·31-s + 1.03·32-s − 0.118·34-s + 0.770·37-s + 0.337·38-s − 0.732·41-s + 1.38·43-s − 0.941·44-s + 0.110·46-s − 1.20·47-s − 0.985·49-s − 0.912·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.337798112\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.337798112\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 0.751T + 2T^{2} \) |
| 7 | \( 1 - 0.315T + 7T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 + 0.917T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 + 7.03T + 29T^{2} \) |
| 31 | \( 1 + 0.867T + 31T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 + 4.69T + 41T^{2} \) |
| 43 | \( 1 - 9.08T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 1.50T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 1.68T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 7.39T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.274630762464333084817522080081, −7.54865041745659419924804329965, −6.45507279200068966833262642815, −6.10686850549352964352884433704, −5.21522090827369918811945083251, −4.48290068893481314542933328191, −3.68305199212705959885236700403, −3.29908452242302398282439554410, −1.81046510508902765838870196391, −0.809447130807835288857627924795,
0.809447130807835288857627924795, 1.81046510508902765838870196391, 3.29908452242302398282439554410, 3.68305199212705959885236700403, 4.48290068893481314542933328191, 5.21522090827369918811945083251, 6.10686850549352964352884433704, 6.45507279200068966833262642815, 7.54865041745659419924804329965, 8.274630762464333084817522080081