| L(s) = 1 | − 1.40·2-s + 1.52·3-s − 0.0258·4-s − 1.01·5-s − 2.14·6-s + 2.90·7-s + 2.84·8-s − 0.662·9-s + 1.42·10-s − 2.94·11-s − 0.0394·12-s − 4.08·14-s − 1.55·15-s − 3.94·16-s + 5.52·17-s + 0.930·18-s + 3.13·19-s + 0.0262·20-s + 4.44·21-s + 4.13·22-s − 3.87·23-s + 4.35·24-s − 3.96·25-s − 5.59·27-s − 0.0750·28-s + 5.59·29-s + 2.18·30-s + ⋯ |
| L(s) = 1 | − 0.993·2-s + 0.882·3-s − 0.0129·4-s − 0.454·5-s − 0.876·6-s + 1.09·7-s + 1.00·8-s − 0.220·9-s + 0.451·10-s − 0.886·11-s − 0.0113·12-s − 1.09·14-s − 0.401·15-s − 0.986·16-s + 1.34·17-s + 0.219·18-s + 0.719·19-s + 0.00586·20-s + 0.970·21-s + 0.881·22-s − 0.808·23-s + 0.888·24-s − 0.793·25-s − 1.07·27-s − 0.0141·28-s + 1.03·29-s + 0.398·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.327800774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327800774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.40T + 2T^{2} \) |
| 3 | \( 1 - 1.52T + 3T^{2} \) |
| 5 | \( 1 + 1.01T + 5T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 19 | \( 1 - 3.13T + 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 - 5.59T + 29T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 - 9.64T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 - 4.26T + 67T^{2} \) |
| 71 | \( 1 - 9.44T + 71T^{2} \) |
| 73 | \( 1 + 0.784T + 73T^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 + 8.81T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 - 6.44T + 97T^{2} \) |
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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
−8.109360636233324700748684746298, −7.67616547347643901803309300633, −7.58548683490416714196642250423, −6.01803551571950518731331523351, −5.22457044642112390790906189486, −4.47180921974360230996032393696, −3.61738404920719033400917811034, −2.67291078318976019793556191418, −1.77241004552266384890744174676, −0.71263327345462643197236142363,
0.71263327345462643197236142363, 1.77241004552266384890744174676, 2.67291078318976019793556191418, 3.61738404920719033400917811034, 4.47180921974360230996032393696, 5.22457044642112390790906189486, 6.01803551571950518731331523351, 7.58548683490416714196642250423, 7.67616547347643901803309300633, 8.109360636233324700748684746298
