L(s) = 1 | − 2.70·2-s + 5.34·4-s − 1.07·7-s − 9.04·8-s − 11-s + 4.34·13-s + 2.92·14-s + 13.8·16-s + 7.75·17-s + 5.26·19-s + 2.70·22-s − 2.15·23-s − 11.7·26-s − 5.75·28-s − 1.41·29-s − 4.68·31-s − 19.3·32-s − 21.0·34-s + 2·37-s − 14.2·38-s + 9.41·41-s − 7.60·43-s − 5.34·44-s + 5.84·46-s + 4.68·47-s − 5.83·49-s + 23.1·52-s + ⋯ |
L(s) = 1 | − 1.91·2-s + 2.67·4-s − 0.407·7-s − 3.19·8-s − 0.301·11-s + 1.20·13-s + 0.780·14-s + 3.45·16-s + 1.88·17-s + 1.20·19-s + 0.577·22-s − 0.449·23-s − 2.30·26-s − 1.08·28-s − 0.263·29-s − 0.840·31-s − 3.42·32-s − 3.60·34-s + 0.328·37-s − 2.31·38-s + 1.47·41-s − 1.15·43-s − 0.805·44-s + 0.861·46-s + 0.682·47-s − 0.833·49-s + 3.21·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7700792340\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7700792340\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 7.75T + 17T^{2} \) |
| 19 | \( 1 - 5.26T + 19T^{2} \) |
| 23 | \( 1 + 2.15T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 9.41T + 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 - 0.156T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 - 8.68T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 8.09T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.061081579155716882766913268351, −8.065425533008553259484504372142, −7.77329965653966197018633728137, −6.94768065661682225446732753205, −6.03986585962032891931000608521, −5.48876852697537827736456793678, −3.60873364680059656527923423887, −2.95132670489929551242914862052, −1.65192556245490859446944456158, −0.78022008218009118041254602973,
0.78022008218009118041254602973, 1.65192556245490859446944456158, 2.95132670489929551242914862052, 3.60873364680059656527923423887, 5.48876852697537827736456793678, 6.03986585962032891931000608521, 6.94768065661682225446732753205, 7.77329965653966197018633728137, 8.065425533008553259484504372142, 9.061081579155716882766913268351