| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s − 18.5·7-s + 8·8-s + 10·10-s − 47.9·11-s + 42.3·13-s − 37.0·14-s + 16·16-s − 1.70·17-s + 21.4·19-s + 20·20-s − 95.8·22-s + 23·23-s + 25·25-s + 84.7·26-s − 74.0·28-s − 57.6·29-s + 295.·31-s + 32·32-s − 3.41·34-s − 92.5·35-s − 7.85·37-s + 42.8·38-s + 40·40-s − 465.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.999·7-s + 0.353·8-s + 0.316·10-s − 1.31·11-s + 0.903·13-s − 0.706·14-s + 0.250·16-s − 0.0243·17-s + 0.258·19-s + 0.223·20-s − 0.928·22-s + 0.208·23-s + 0.200·25-s + 0.639·26-s − 0.499·28-s − 0.369·29-s + 1.71·31-s + 0.176·32-s − 0.0172·34-s − 0.446·35-s − 0.0348·37-s + 0.182·38-s + 0.158·40-s − 1.77·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.177991785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.177991785\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 + 18.5T + 343T^{2} \) |
| 11 | \( 1 + 47.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 1.70T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 57.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 7.85T + 5.06e4T^{2} \) |
| 41 | \( 1 + 465.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 449.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 368.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 377.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 849.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 92.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 626.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 439.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 641.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3T + 9.12e5T^{2} \) |
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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
−8.651727894030545101978550966513, −7.996365657287612146796835646962, −6.90062763856064692564063746151, −6.37269538949155017563912160295, −5.53297577015500989333153686960, −4.88828795539854754750714780675, −3.68770339208868477079726375899, −3.00382639881459934394182280223, −2.11124055181872467287244434321, −0.71337618386922101725451085150,
0.71337618386922101725451085150, 2.11124055181872467287244434321, 3.00382639881459934394182280223, 3.68770339208868477079726375899, 4.88828795539854754750714780675, 5.53297577015500989333153686960, 6.37269538949155017563912160295, 6.90062763856064692564063746151, 7.996365657287612146796835646962, 8.651727894030545101978550966513
