L(s) = 1 | + 2-s + 4-s + 3.02·5-s − 3.77·7-s + 8-s + 3.02·10-s − 1.20·11-s − 1.04·13-s − 3.77·14-s + 16-s − 3.48·17-s − 4.38·19-s + 3.02·20-s − 1.20·22-s − 6.93·23-s + 4.12·25-s − 1.04·26-s − 3.77·28-s + 1.44·29-s − 3.28·31-s + 32-s − 3.48·34-s − 11.3·35-s − 5.93·37-s − 4.38·38-s + 3.02·40-s − 5.58·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.35·5-s − 1.42·7-s + 0.353·8-s + 0.955·10-s − 0.362·11-s − 0.290·13-s − 1.00·14-s + 0.250·16-s − 0.844·17-s − 1.00·19-s + 0.675·20-s − 0.256·22-s − 1.44·23-s + 0.824·25-s − 0.205·26-s − 0.712·28-s + 0.268·29-s − 0.590·31-s + 0.176·32-s − 0.597·34-s − 1.92·35-s − 0.975·37-s − 0.712·38-s + 0.477·40-s − 0.872·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 5 | \( 1 - 3.02T + 5T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 + 1.20T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 + 3.48T + 17T^{2} \) |
| 19 | \( 1 + 4.38T + 19T^{2} \) |
| 23 | \( 1 + 6.93T + 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 - 9.61T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 7.69T + 67T^{2} \) |
| 71 | \( 1 + 2.66T + 71T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 79 | \( 1 + 7.37T + 79T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 - 6.47T + 89T^{2} \) |
| 97 | \( 1 - 8.21T + 97T^{2} \) |
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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
−8.027900632040529443544022340819, −6.97850066772823858231853284767, −6.37435295377596256897870516292, −5.99681506848008122852417763663, −5.20215737038215989748935838975, −4.26765570798430688114115023466, −3.37955323140548571450300994581, −2.45222773144984568089470746235, −1.88009574471048767165434544330, 0,
1.88009574471048767165434544330, 2.45222773144984568089470746235, 3.37955323140548571450300994581, 4.26765570798430688114115023466, 5.20215737038215989748935838975, 5.99681506848008122852417763663, 6.37435295377596256897870516292, 6.97850066772823858231853284767, 8.027900632040529443544022340819