| L(s) = 1 | − 1.04·2-s + 2.85·3-s − 0.898·4-s + 1.97·5-s − 2.99·6-s + 2.56·7-s + 3.04·8-s + 5.16·9-s − 2.07·10-s − 1.89·11-s − 2.56·12-s + 13-s − 2.69·14-s + 5.65·15-s − 1.39·16-s + 2.38·17-s − 5.42·18-s − 0.230·19-s − 1.77·20-s + 7.33·21-s + 1.99·22-s − 3.31·23-s + 8.69·24-s − 1.08·25-s − 1.04·26-s + 6.18·27-s − 2.30·28-s + ⋯ |
| L(s) = 1 | − 0.742·2-s + 1.64·3-s − 0.449·4-s + 0.884·5-s − 1.22·6-s + 0.969·7-s + 1.07·8-s + 1.72·9-s − 0.656·10-s − 0.572·11-s − 0.741·12-s + 0.277·13-s − 0.719·14-s + 1.45·15-s − 0.349·16-s + 0.579·17-s − 1.27·18-s − 0.0528·19-s − 0.397·20-s + 1.59·21-s + 0.424·22-s − 0.691·23-s + 1.77·24-s − 0.217·25-s − 0.205·26-s + 1.19·27-s − 0.435·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.347869067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.347869067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 3 | \( 1 - 2.85T + 3T^{2} \) |
| 5 | \( 1 - 1.97T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 1.89T + 11T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 + 0.230T + 19T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 - 9.36T + 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 + 9.87T + 37T^{2} \) |
| 41 | \( 1 - 1.98T + 41T^{2} \) |
| 43 | \( 1 + 2.61T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 + 0.691T + 59T^{2} \) |
| 61 | \( 1 - 6.46T + 61T^{2} \) |
| 67 | \( 1 - 1.83T + 67T^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 9.74T + 79T^{2} \) |
| 83 | \( 1 - 4.95T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−9.648218304125523956744298986746, −8.579313502070274180335906295664, −8.253425982663688456742693239616, −7.76830880776952988073774872357, −6.59151956434717987158094113490, −5.22028839446944291324548002796, −4.45600446416546157308171587679, −3.29126777173007460978854005917, −2.16738351236704038437205027184, −1.36859398256722082521561361335,
1.36859398256722082521561361335, 2.16738351236704038437205027184, 3.29126777173007460978854005917, 4.45600446416546157308171587679, 5.22028839446944291324548002796, 6.59151956434717987158094113490, 7.76830880776952988073774872357, 8.253425982663688456742693239616, 8.579313502070274180335906295664, 9.648218304125523956744298986746
