| L(s) = 1 | + 2.20·2-s − 2.28·3-s + 2.84·4-s + 5-s − 5.03·6-s − 0.723·7-s + 1.85·8-s + 2.23·9-s + 2.20·10-s − 0.340·11-s − 6.50·12-s − 0.0740·13-s − 1.59·14-s − 2.28·15-s − 1.60·16-s + 4.05·17-s + 4.92·18-s + 3.30·19-s + 2.84·20-s + 1.65·21-s − 0.749·22-s + 4.02·23-s − 4.24·24-s + 25-s − 0.163·26-s + 1.74·27-s − 2.05·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s − 1.32·3-s + 1.42·4-s + 0.447·5-s − 2.05·6-s − 0.273·7-s + 0.656·8-s + 0.745·9-s + 0.695·10-s − 0.102·11-s − 1.87·12-s − 0.0205·13-s − 0.425·14-s − 0.590·15-s − 0.400·16-s + 0.984·17-s + 1.16·18-s + 0.758·19-s + 0.635·20-s + 0.361·21-s − 0.159·22-s + 0.839·23-s − 0.867·24-s + 0.200·25-s − 0.0319·26-s + 0.336·27-s − 0.388·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.849345438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.849345438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 3 | \( 1 + 2.28T + 3T^{2} \) |
| 7 | \( 1 + 0.723T + 7T^{2} \) |
| 11 | \( 1 + 0.340T + 11T^{2} \) |
| 13 | \( 1 + 0.0740T + 13T^{2} \) |
| 17 | \( 1 - 4.05T + 17T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 23 | \( 1 - 4.02T + 23T^{2} \) |
| 29 | \( 1 - 5.39T + 29T^{2} \) |
| 31 | \( 1 - 3.92T + 31T^{2} \) |
| 37 | \( 1 - 2.07T + 37T^{2} \) |
| 41 | \( 1 + 6.08T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 - 0.959T + 47T^{2} \) |
| 53 | \( 1 - 0.920T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 4.61T + 61T^{2} \) |
| 67 | \( 1 - 5.45T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 8.35T + 73T^{2} \) |
| 79 | \( 1 - 4.54T + 79T^{2} \) |
| 83 | \( 1 + 7.41T + 83T^{2} \) |
| 89 | \( 1 + 1.47T + 89T^{2} \) |
| 97 | \( 1 + 6.56T + 97T^{2} \) |
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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
−9.423900813210654304593690046537, −8.189915452562441408799744373576, −6.97158698067900700800977067333, −6.51111893122539270927314868608, −5.67248256962580534870442563316, −5.26306962755554113789960514057, −4.57066052507095868762558750446, −3.45599782719110930641358107974, −2.60254548385298905090198981238, −0.983745589881359003487753995040,
0.983745589881359003487753995040, 2.60254548385298905090198981238, 3.45599782719110930641358107974, 4.57066052507095868762558750446, 5.26306962755554113789960514057, 5.67248256962580534870442563316, 6.51111893122539270927314868608, 6.97158698067900700800977067333, 8.189915452562441408799744373576, 9.423900813210654304593690046537
