L(s) = 1 | + 2-s − 2.76·3-s + 4-s + 4.07·5-s − 2.76·6-s + 2.61·7-s + 8-s + 4.66·9-s + 4.07·10-s − 4.42·11-s − 2.76·12-s − 6.42·13-s + 2.61·14-s − 11.2·15-s + 16-s − 4.31·17-s + 4.66·18-s − 3.26·19-s + 4.07·20-s − 7.24·21-s − 4.42·22-s + 3.62·23-s − 2.76·24-s + 11.6·25-s − 6.42·26-s − 4.60·27-s + 2.61·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.59·3-s + 0.5·4-s + 1.82·5-s − 1.13·6-s + 0.989·7-s + 0.353·8-s + 1.55·9-s + 1.28·10-s − 1.33·11-s − 0.799·12-s − 1.78·13-s + 0.699·14-s − 2.91·15-s + 0.250·16-s − 1.04·17-s + 1.09·18-s − 0.748·19-s + 0.911·20-s − 1.58·21-s − 0.943·22-s + 0.755·23-s − 0.565·24-s + 2.32·25-s − 1.25·26-s − 0.885·27-s + 0.494·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 4001 | \( 1+O(T) \) |
good | 3 | \( 1 + 2.76T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 + 4.42T + 11T^{2} \) |
| 13 | \( 1 + 6.42T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 + 1.10T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 1.48T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 + 8.55T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 6.07T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 - 3.42T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 - 0.250T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.08367349678861348407097613455, −6.52368360607952650422521806819, −6.01347236680435850650389022222, −5.13526570395820117501267076497, −4.91799999649628774577825459762, −4.67322932284053069921435014903, −2.78918721800251249635214510520, −2.22340670172562746361305451741, −1.44575417452835789177766471701, 0,
1.44575417452835789177766471701, 2.22340670172562746361305451741, 2.78918721800251249635214510520, 4.67322932284053069921435014903, 4.91799999649628774577825459762, 5.13526570395820117501267076497, 6.01347236680435850650389022222, 6.52368360607952650422521806819, 7.08367349678861348407097613455