L(s) = 1 | − 2.41·2-s + 3-s + 3.81·4-s − 0.0291·5-s − 2.41·6-s − 4.28·7-s − 4.36·8-s + 9-s + 0.0703·10-s − 2.79·11-s + 3.81·12-s − 0.529·13-s + 10.3·14-s − 0.0291·15-s + 2.90·16-s − 6.45·17-s − 2.41·18-s + 3.36·19-s − 0.111·20-s − 4.28·21-s + 6.74·22-s + 1.83·23-s − 4.36·24-s − 4.99·25-s + 1.27·26-s + 27-s − 16.3·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 0.577·3-s + 1.90·4-s − 0.0130·5-s − 0.984·6-s − 1.61·7-s − 1.54·8-s + 0.333·9-s + 0.0222·10-s − 0.843·11-s + 1.10·12-s − 0.146·13-s + 2.75·14-s − 0.00753·15-s + 0.725·16-s − 1.56·17-s − 0.568·18-s + 0.771·19-s − 0.0248·20-s − 0.934·21-s + 1.43·22-s + 0.383·23-s − 0.891·24-s − 0.999·25-s + 0.250·26-s + 0.192·27-s − 3.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3234351831\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3234351831\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 2671 | \( 1 + T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 0.0291T + 5T^{2} \) |
| 7 | \( 1 + 4.28T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 0.529T + 13T^{2} \) |
| 17 | \( 1 + 6.45T + 17T^{2} \) |
| 19 | \( 1 - 3.36T + 19T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 - 6.02T + 37T^{2} \) |
| 41 | \( 1 + 5.62T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 + 0.386T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 - 8.57T + 73T^{2} \) |
| 79 | \( 1 + 4.68T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.952153853216702525393681851062, −7.35391335856960570909904373970, −6.65300754277505296700593335628, −6.29214950592189371463406788976, −5.13974679834536540504626955594, −4.06022723356080817249778262063, −3.00434173979669826887706592657, −2.62277396568025992534803590721, −1.64122281394625942193591962983, −0.34093897010037897756367525399,
0.34093897010037897756367525399, 1.64122281394625942193591962983, 2.62277396568025992534803590721, 3.00434173979669826887706592657, 4.06022723356080817249778262063, 5.13974679834536540504626955594, 6.29214950592189371463406788976, 6.65300754277505296700593335628, 7.35391335856960570909904373970, 7.952153853216702525393681851062