L(s) = 1 | − 2.43·2-s − 3.01·3-s + 3.93·4-s + 2.61·5-s + 7.35·6-s − 1.67·7-s − 4.71·8-s + 6.11·9-s − 6.36·10-s − 3.15·11-s − 11.8·12-s + 5.74·13-s + 4.07·14-s − 7.88·15-s + 3.62·16-s − 4.74·17-s − 14.9·18-s + 0.791·19-s + 10.2·20-s + 5.04·21-s + 7.69·22-s + 23-s + 14.2·24-s + 1.82·25-s − 13.9·26-s − 9.41·27-s − 6.57·28-s + ⋯ |
L(s) = 1 | − 1.72·2-s − 1.74·3-s + 1.96·4-s + 1.16·5-s + 3.00·6-s − 0.631·7-s − 1.66·8-s + 2.03·9-s − 2.01·10-s − 0.952·11-s − 3.43·12-s + 1.59·13-s + 1.08·14-s − 2.03·15-s + 0.905·16-s − 1.15·17-s − 3.51·18-s + 0.181·19-s + 2.29·20-s + 1.10·21-s + 1.64·22-s + 0.208·23-s + 2.90·24-s + 0.364·25-s − 2.74·26-s − 1.81·27-s − 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2650395398\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2650395398\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 - T \) |
| 349 | \( 1 + T \) |
good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 + 3.01T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 - 5.74T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 0.791T + 19T^{2} \) |
| 29 | \( 1 + 0.131T + 29T^{2} \) |
| 31 | \( 1 + 9.74T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 + 8.38T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 - 5.04T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 6.91T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 3.41T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 5.66T + 89T^{2} \) |
| 97 | \( 1 - 5.87T + 97T^{2} \) |
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
−7.81267905966936383995180986389, −6.91811057736450723360151804261, −6.62904446316513281141378748949, −5.84380686514770728896796963594, −5.58977863281633330874990662332, −4.54327277414806082855360556782, −3.27100154675365847429474889475, −2.01949880557688024796739929139, −1.46456752266190159943153593964, −0.37752505962998130067727934774,
0.37752505962998130067727934774, 1.46456752266190159943153593964, 2.01949880557688024796739929139, 3.27100154675365847429474889475, 4.54327277414806082855360556782, 5.58977863281633330874990662332, 5.84380686514770728896796963594, 6.62904446316513281141378748949, 6.91811057736450723360151804261, 7.81267905966936383995180986389