| L(s) = 1 | + 2-s + 4-s + 1.39·5-s + 4.29·7-s + 8-s + 1.39·10-s + 0.221·11-s + 1.98·13-s + 4.29·14-s + 16-s + 0.111·17-s − 3.23·19-s + 1.39·20-s + 0.221·22-s − 3.04·25-s + 1.98·26-s + 4.29·28-s + 3.56·29-s + 6.15·31-s + 32-s + 0.111·34-s + 5.99·35-s − 4.22·37-s − 3.23·38-s + 1.39·40-s − 3.41·41-s + 3.24·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.625·5-s + 1.62·7-s + 0.353·8-s + 0.442·10-s + 0.0668·11-s + 0.551·13-s + 1.14·14-s + 0.250·16-s + 0.0269·17-s − 0.742·19-s + 0.312·20-s + 0.0472·22-s − 0.609·25-s + 0.389·26-s + 0.811·28-s + 0.662·29-s + 1.10·31-s + 0.176·32-s + 0.0190·34-s + 1.01·35-s − 0.695·37-s − 0.524·38-s + 0.221·40-s − 0.533·41-s + 0.495·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.188375076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.188375076\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 1.39T + 5T^{2} \) |
| 7 | \( 1 - 4.29T + 7T^{2} \) |
| 11 | \( 1 - 0.221T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 - 0.111T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 - 6.15T + 31T^{2} \) |
| 37 | \( 1 + 4.22T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 - 0.146T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 5.84T + 71T^{2} \) |
| 73 | \( 1 + 3.79T + 73T^{2} \) |
| 79 | \( 1 - 8.72T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 1.01T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−7.74224068088448364164195591412, −6.80163478960123000976642514551, −6.29444236665101635872148333159, −5.45943972532074682545854236159, −5.00240407674857797210838404910, −4.29967641244450182736485045594, −3.60507267711771499709861079542, −2.45058851639707065097562778765, −1.89420058538581240531784208102, −1.04993233430508583197038937213,
1.04993233430508583197038937213, 1.89420058538581240531784208102, 2.45058851639707065097562778765, 3.60507267711771499709861079542, 4.29967641244450182736485045594, 5.00240407674857797210838404910, 5.45943972532074682545854236159, 6.29444236665101635872148333159, 6.80163478960123000976642514551, 7.74224068088448364164195591412
