L(s) = 1 | − 2.33·2-s + 2.29·3-s + 3.45·4-s − 0.314·5-s − 5.35·6-s + 0.543·7-s − 3.40·8-s + 2.24·9-s + 0.734·10-s + 11-s + 7.92·12-s + 6.37·13-s − 1.26·14-s − 0.720·15-s + 1.04·16-s + 7.80·17-s − 5.24·18-s − 4.51·19-s − 1.08·20-s + 1.24·21-s − 2.33·22-s + 3.63·23-s − 7.80·24-s − 4.90·25-s − 14.8·26-s − 1.72·27-s + 1.87·28-s + ⋯ |
L(s) = 1 | − 1.65·2-s + 1.32·3-s + 1.72·4-s − 0.140·5-s − 2.18·6-s + 0.205·7-s − 1.20·8-s + 0.749·9-s + 0.232·10-s + 0.301·11-s + 2.28·12-s + 1.76·13-s − 0.339·14-s − 0.186·15-s + 0.260·16-s + 1.89·17-s − 1.23·18-s − 1.03·19-s − 0.243·20-s + 0.271·21-s − 0.498·22-s + 0.758·23-s − 1.59·24-s − 0.980·25-s − 2.92·26-s − 0.331·27-s + 0.354·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.367218276\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.367218276\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 3 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 + 0.314T + 5T^{2} \) |
| 7 | \( 1 - 0.543T + 7T^{2} \) |
| 13 | \( 1 - 6.37T + 13T^{2} \) |
| 17 | \( 1 - 7.80T + 17T^{2} \) |
| 19 | \( 1 + 4.51T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 - 0.536T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 + 3.76T + 53T^{2} \) |
| 59 | \( 1 - 9.78T + 59T^{2} \) |
| 61 | \( 1 + 4.79T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 8.17T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 2.12T + 83T^{2} \) |
| 89 | \( 1 + 7.45T + 89T^{2} \) |
| 97 | \( 1 + 0.288T + 97T^{2} \) |
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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
−9.390254116661636295949028808821, −8.583186298637002546630324287456, −8.227498035520460964261063595131, −7.63729249779889775949597118005, −6.68638258586148817346182479848, −5.69356053304154733122829053031, −3.97870981613340722514393329236, −3.21921436829981635347723753017, −2.01179864820729791707353630678, −1.09382864833251643783732868469,
1.09382864833251643783732868469, 2.01179864820729791707353630678, 3.21921436829981635347723753017, 3.97870981613340722514393329236, 5.69356053304154733122829053031, 6.68638258586148817346182479848, 7.63729249779889775949597118005, 8.227498035520460964261063595131, 8.583186298637002546630324287456, 9.390254116661636295949028808821