L(s) = 1 | − 2.15·2-s − 0.0701·3-s + 2.66·4-s − 2.25·5-s + 0.151·6-s − 1.38·7-s − 1.42·8-s − 2.99·9-s + 4.86·10-s − 4.35·11-s − 0.186·12-s − 13-s + 2.98·14-s + 0.158·15-s − 2.24·16-s + 3.87·17-s + 6.46·18-s − 7.82·19-s − 5.99·20-s + 0.0968·21-s + 9.40·22-s − 2.49·23-s + 0.100·24-s + 0.0834·25-s + 2.15·26-s + 0.420·27-s − 3.67·28-s + ⋯ |
L(s) = 1 | − 1.52·2-s − 0.0404·3-s + 1.33·4-s − 1.00·5-s + 0.0617·6-s − 0.521·7-s − 0.504·8-s − 0.998·9-s + 1.53·10-s − 1.31·11-s − 0.0538·12-s − 0.277·13-s + 0.796·14-s + 0.0408·15-s − 0.560·16-s + 0.939·17-s + 1.52·18-s − 1.79·19-s − 1.34·20-s + 0.0211·21-s + 2.00·22-s − 0.520·23-s + 0.0204·24-s + 0.0166·25-s + 0.423·26-s + 0.0808·27-s − 0.694·28-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
\(L(1)\) |
\(\approx\) |
\(0.1608929794\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1608929794\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 + 0.0701T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 + 7.82T + 19T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 - 6.85T + 29T^{2} \) |
| 31 | \( 1 + 7.84T + 31T^{2} \) |
| 37 | \( 1 - 0.413T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 6.57T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 1.00T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 - 0.796T + 79T^{2} \) |
| 83 | \( 1 - 1.36T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 5.58T + 97T^{2} \) |
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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
−9.613919444834834909526405737037, −8.579673145249343872589895059990, −8.158530206430610793426845339795, −7.59767238990658814082395571489, −6.64182186364225607303927122468, −5.65638918067651378412742017855, −4.44818341197338425526740713568, −3.19151388887531725237173614132, −2.20666340022641749508331071524, −0.34153267758274259013425016831,
0.34153267758274259013425016831, 2.20666340022641749508331071524, 3.19151388887531725237173614132, 4.44818341197338425526740713568, 5.65638918067651378412742017855, 6.64182186364225607303927122468, 7.59767238990658814082395571489, 8.158530206430610793426845339795, 8.579673145249343872589895059990, 9.613919444834834909526405737037