L(s) = 1 | − 2-s + 3.22·3-s + 4-s + 3.83·5-s − 3.22·6-s − 1.13·7-s − 8-s + 7.38·9-s − 3.83·10-s − 11-s + 3.22·12-s + 5.72·13-s + 1.13·14-s + 12.3·15-s + 16-s − 3.45·17-s − 7.38·18-s + 3.83·20-s − 3.65·21-s + 22-s + 3.83·23-s − 3.22·24-s + 9.74·25-s − 5.72·26-s + 14.1·27-s − 1.13·28-s − 1.43·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.86·3-s + 0.5·4-s + 1.71·5-s − 1.31·6-s − 0.428·7-s − 0.353·8-s + 2.46·9-s − 1.21·10-s − 0.301·11-s + 0.930·12-s + 1.58·13-s + 0.303·14-s + 3.19·15-s + 0.250·16-s − 0.837·17-s − 1.73·18-s + 0.858·20-s − 0.797·21-s + 0.213·22-s + 0.800·23-s − 0.657·24-s + 1.94·25-s − 1.12·26-s + 2.71·27-s − 0.214·28-s − 0.266·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.669410386\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.669410386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 3.22T + 3T^{2} \) |
| 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 13 | \( 1 - 5.72T + 13T^{2} \) |
| 17 | \( 1 + 3.45T + 17T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 + 1.43T + 29T^{2} \) |
| 31 | \( 1 - 0.978T + 31T^{2} \) |
| 37 | \( 1 + 8.43T + 37T^{2} \) |
| 41 | \( 1 + 0.839T + 41T^{2} \) |
| 43 | \( 1 - 2.67T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 1.66T + 67T^{2} \) |
| 71 | \( 1 - 9.98T + 71T^{2} \) |
| 73 | \( 1 + 5.76T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 7.15T + 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−8.309854176626161845269545048796, −7.02344561300221996974433523440, −6.82455357993884331335309486195, −5.96482608948430992255517885078, −5.11915443702258994279890437060, −3.93939013012842051134079965422, −3.20104026948268111396799534387, −2.52536120076818578992151134687, −1.85875542195603187290474445371, −1.21032489753899351961314220122,
1.21032489753899351961314220122, 1.85875542195603187290474445371, 2.52536120076818578992151134687, 3.20104026948268111396799534387, 3.93939013012842051134079965422, 5.11915443702258994279890437060, 5.96482608948430992255517885078, 6.82455357993884331335309486195, 7.02344561300221996974433523440, 8.309854176626161845269545048796