| L(s) = 1 | − 2-s + 0.569·3-s + 4-s − 3.97·5-s − 0.569·6-s + 1.73·7-s − 8-s − 2.67·9-s + 3.97·10-s − 2.32·11-s + 0.569·12-s − 2.81·13-s − 1.73·14-s − 2.26·15-s + 16-s − 5.96·17-s + 2.67·18-s − 0.871·19-s − 3.97·20-s + 0.987·21-s + 2.32·22-s − 6.37·23-s − 0.569·24-s + 10.8·25-s + 2.81·26-s − 3.23·27-s + 1.73·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.328·3-s + 0.5·4-s − 1.77·5-s − 0.232·6-s + 0.655·7-s − 0.353·8-s − 0.891·9-s + 1.25·10-s − 0.701·11-s + 0.164·12-s − 0.780·13-s − 0.463·14-s − 0.584·15-s + 0.250·16-s − 1.44·17-s + 0.630·18-s − 0.199·19-s − 0.888·20-s + 0.215·21-s + 0.495·22-s − 1.32·23-s − 0.116·24-s + 2.16·25-s + 0.552·26-s − 0.622·27-s + 0.327·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2796451576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2796451576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 0.569T + 3T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 2.32T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 + 0.871T + 19T^{2} \) |
| 23 | \( 1 + 6.37T + 23T^{2} \) |
| 29 | \( 1 + 3.28T + 29T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 - 0.929T + 37T^{2} \) |
| 41 | \( 1 - 8.86T + 41T^{2} \) |
| 43 | \( 1 - 0.761T + 43T^{2} \) |
| 47 | \( 1 - 0.352T + 47T^{2} \) |
| 53 | \( 1 - 2.86T + 53T^{2} \) |
| 59 | \( 1 + 8.31T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 + 0.289T + 67T^{2} \) |
| 71 | \( 1 - 4.17T + 71T^{2} \) |
| 73 | \( 1 + 1.95T + 73T^{2} \) |
| 79 | \( 1 - 0.582T + 79T^{2} \) |
| 83 | \( 1 - 4.25T + 83T^{2} \) |
| 89 | \( 1 + 7.72T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−8.383246681176981494358109156235, −7.72377809756884076087369014838, −7.49195377221604357028295383319, −6.47444454320216492722937080828, −5.42691733143088643613966196506, −4.51019215496165602743798890107, −3.84662517495904051034613506349, −2.80355727364074630024194855249, −2.06386730682802245212097534327, −0.30769466779547539224605109167,
0.30769466779547539224605109167, 2.06386730682802245212097534327, 2.80355727364074630024194855249, 3.84662517495904051034613506349, 4.51019215496165602743798890107, 5.42691733143088643613966196506, 6.47444454320216492722937080828, 7.49195377221604357028295383319, 7.72377809756884076087369014838, 8.383246681176981494358109156235
