L(s) = 1 | − 0.444·2-s − 1.68·3-s − 1.80·4-s + 3.41·5-s + 0.749·6-s − 1.25·7-s + 1.69·8-s − 0.157·9-s − 1.51·10-s + 3.03·12-s + 4.74·13-s + 0.558·14-s − 5.76·15-s + 2.85·16-s − 6.97·17-s + 0.0697·18-s + 5.14·19-s − 6.15·20-s + 2.11·21-s − 6.04·23-s − 2.84·24-s + 6.67·25-s − 2.10·26-s + 5.32·27-s + 2.26·28-s − 1.09·29-s + 2.56·30-s + ⋯ |
L(s) = 1 | − 0.314·2-s − 0.973·3-s − 0.901·4-s + 1.52·5-s + 0.305·6-s − 0.474·7-s + 0.597·8-s − 0.0523·9-s − 0.480·10-s + 0.877·12-s + 1.31·13-s + 0.149·14-s − 1.48·15-s + 0.713·16-s − 1.69·17-s + 0.0164·18-s + 1.17·19-s − 1.37·20-s + 0.462·21-s − 1.26·23-s − 0.581·24-s + 1.33·25-s − 0.413·26-s + 1.02·27-s + 0.427·28-s − 0.203·29-s + 0.467·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.028092705\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.028092705\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 0.444T + 2T^{2} \) |
| 3 | \( 1 + 1.68T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 + 6.97T + 17T^{2} \) |
| 19 | \( 1 - 5.14T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 + 6.96T + 47T^{2} \) |
| 53 | \( 1 - 6.62T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 8.58T + 73T^{2} \) |
| 79 | \( 1 + 0.574T + 79T^{2} \) |
| 83 | \( 1 + 9.44T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−8.146984215193201485260933904971, −6.89283045828113386859522527231, −6.38585176964329145342600759171, −5.70477839323983917827937567872, −5.40177520594719145905751506171, −4.48220216745879316326307974794, −3.68124906697889974297693187301, −2.54136115678929442118512132898, −1.53712025111183228802447337262, −0.59587009204736348747211025803,
0.59587009204736348747211025803, 1.53712025111183228802447337262, 2.54136115678929442118512132898, 3.68124906697889974297693187301, 4.48220216745879316326307974794, 5.40177520594719145905751506171, 5.70477839323983917827937567872, 6.38585176964329145342600759171, 6.89283045828113386859522527231, 8.146984215193201485260933904971