| L(s) = 1 | + 0.381·2-s − 3-s − 1.85·4-s − 0.381·6-s + 3.61·7-s − 1.47·8-s + 9-s − 1.76·11-s + 1.85·12-s + 3·13-s + 1.38·14-s + 3.14·16-s + 5.61·17-s + 0.381·18-s − 19-s − 3.61·21-s − 0.673·22-s + 6.70·23-s + 1.47·24-s + 1.14·26-s − 27-s − 6.70·28-s + 0.236·29-s + 8.09·31-s + 4.14·32-s + 1.76·33-s + 2.14·34-s + ⋯ |
| L(s) = 1 | + 0.270·2-s − 0.577·3-s − 0.927·4-s − 0.155·6-s + 1.36·7-s − 0.520·8-s + 0.333·9-s − 0.531·11-s + 0.535·12-s + 0.832·13-s + 0.369·14-s + 0.786·16-s + 1.36·17-s + 0.0900·18-s − 0.229·19-s − 0.789·21-s − 0.143·22-s + 1.39·23-s + 0.300·24-s + 0.224·26-s − 0.192·27-s − 1.26·28-s + 0.0438·29-s + 1.45·31-s + 0.732·32-s + 0.307·33-s + 0.368·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224261952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224261952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 - 0.236T + 29T^{2} \) |
| 31 | \( 1 - 8.09T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 4.61T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 6.09T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 7.85T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 - 7.32T + 83T^{2} \) |
| 89 | \( 1 - 3.47T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−11.42056186132373665349362720418, −10.56946246442556154689296123139, −9.633586279559574457980072021446, −8.392883339219356181089538172669, −7.908872451737295847062049014185, −6.38078921596589765378089247921, −5.17684851707946292463659509748, −4.78435399131616864939931411152, −3.36832262584671559576648608805, −1.20724378145211470931103074198,
1.20724378145211470931103074198, 3.36832262584671559576648608805, 4.78435399131616864939931411152, 5.17684851707946292463659509748, 6.38078921596589765378089247921, 7.908872451737295847062049014185, 8.392883339219356181089538172669, 9.633586279559574457980072021446, 10.56946246442556154689296123139, 11.42056186132373665349362720418
