L(s) = 1 | + 2.61·2-s − 2.61·3-s + 4.85·4-s + 2.61·5-s − 6.85·6-s + 7-s + 7.47·8-s + 3.85·9-s + 6.85·10-s + 1.85·11-s − 12.7·12-s + 2.61·14-s − 6.85·15-s + 9.85·16-s − 1.47·17-s + 10.0·18-s − 1.85·19-s + 12.7·20-s − 2.61·21-s + 4.85·22-s − 4.47·23-s − 19.5·24-s + 1.85·25-s − 2.23·27-s + 4.85·28-s + 7.09·29-s − 17.9·30-s + ⋯ |
L(s) = 1 | + 1.85·2-s − 1.51·3-s + 2.42·4-s + 1.17·5-s − 2.79·6-s + 0.377·7-s + 2.64·8-s + 1.28·9-s + 2.16·10-s + 0.559·11-s − 3.66·12-s + 0.699·14-s − 1.76·15-s + 2.46·16-s − 0.357·17-s + 2.37·18-s − 0.425·19-s + 2.84·20-s − 0.571·21-s + 1.03·22-s − 0.932·23-s − 3.99·24-s + 0.370·25-s − 0.430·27-s + 0.917·28-s + 1.31·29-s − 3.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.082377603\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.082377603\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 0.763T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 - 3.76T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 - 4.90T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−10.39469235113878481840110883402, −9.159245924366192029806963170055, −7.57891971778310487320048274066, −6.55638763103508474136832820972, −6.10061786304037931337812718176, −5.60629874154245458866093246394, −4.73048112192876361752133074913, −4.10077957951623286033575290346, −2.55842159840189424469881316217, −1.49503934372324026440084297395,
1.49503934372324026440084297395, 2.55842159840189424469881316217, 4.10077957951623286033575290346, 4.73048112192876361752133074913, 5.60629874154245458866093246394, 6.10061786304037931337812718176, 6.55638763103508474136832820972, 7.57891971778310487320048274066, 9.159245924366192029806963170055, 10.39469235113878481840110883402