| L(s) = 1 | − 1.53·2-s + 0.369·4-s + 3.87·7-s + 2.51·8-s − 1.24·11-s − 13-s − 5.97·14-s − 4.60·16-s − 0.659·17-s + 6.97·19-s + 1.92·22-s − 1.55·23-s + 1.53·26-s + 1.43·28-s + 3·29-s + 5.43·31-s + 2.06·32-s + 1.01·34-s + 2.29·37-s − 10.7·38-s + 10.2·41-s − 8.20·43-s − 0.460·44-s + 2.38·46-s − 1.53·47-s + 8.04·49-s − 0.369·52-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.184·4-s + 1.46·7-s + 0.887·8-s − 0.376·11-s − 0.277·13-s − 1.59·14-s − 1.15·16-s − 0.160·17-s + 1.59·19-s + 0.409·22-s − 0.323·23-s + 0.301·26-s + 0.270·28-s + 0.557·29-s + 0.975·31-s + 0.364·32-s + 0.174·34-s + 0.376·37-s − 1.74·38-s + 1.59·41-s − 1.25·43-s − 0.0694·44-s + 0.352·46-s − 0.224·47-s + 1.14·49-s − 0.0511·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.164128169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.164128169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 17 | \( 1 + 0.659T + 17T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 - 2.29T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 8.20T + 43T^{2} \) |
| 47 | \( 1 + 1.53T + 47T^{2} \) |
| 53 | \( 1 - 1.44T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 9.92T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 6.20T + 73T^{2} \) |
| 79 | \( 1 + 0.474T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
| 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
−8.705093682414337767804156555006, −7.928119052755867001310692585123, −7.72407976921478539087948957927, −6.81485633006243485557498146555, −5.57098635249992168778361785562, −4.87149557501528024107760970136, −4.22715090056149202413335508302, −2.80535245679943691204272985323, −1.71915025451487959130131985777, −0.837395647165703164905880332975,
0.837395647165703164905880332975, 1.71915025451487959130131985777, 2.80535245679943691204272985323, 4.22715090056149202413335508302, 4.87149557501528024107760970136, 5.57098635249992168778361785562, 6.81485633006243485557498146555, 7.72407976921478539087948957927, 7.928119052755867001310692585123, 8.705093682414337767804156555006
