| L(s) = 1 | + 2-s + 3-s + 4-s + 3.41·5-s + 6-s + 7-s + 8-s + 9-s + 3.41·10-s + 3.24·11-s + 12-s + 14-s + 3.41·15-s + 16-s + 1.80·17-s + 18-s + 1.93·19-s + 3.41·20-s + 21-s + 3.24·22-s + 3.28·23-s + 24-s + 6.66·25-s + 27-s + 28-s − 3.64·29-s + 3.41·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.52·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.07·10-s + 0.979·11-s + 0.288·12-s + 0.267·14-s + 0.881·15-s + 0.250·16-s + 0.437·17-s + 0.235·18-s + 0.444·19-s + 0.763·20-s + 0.218·21-s + 0.692·22-s + 0.684·23-s + 0.204·24-s + 1.33·25-s + 0.192·27-s + 0.188·28-s − 0.676·29-s + 0.623·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.590988467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.590988467\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 17 | \( 1 - 1.80T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 - 3.28T + 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 + 3.10T + 31T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2.46T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 2.66T + 71T^{2} \) |
| 73 | \( 1 - 0.368T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 1.43T + 83T^{2} \) |
| 89 | \( 1 - 5.10T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.83870858996926130227690509142, −7.03170968986719522032694709946, −6.53036717317170281233202870048, −5.65917604518962914406145226249, −5.25994906059890082768679785977, −4.35861133349349395550136386555, −3.49265402248080159563195502389, −2.77440743131321516552520418532, −1.80166906995486746104578352656, −1.35622863921516371686782223182,
1.35622863921516371686782223182, 1.80166906995486746104578352656, 2.77440743131321516552520418532, 3.49265402248080159563195502389, 4.35861133349349395550136386555, 5.25994906059890082768679785977, 5.65917604518962914406145226249, 6.53036717317170281233202870048, 7.03170968986719522032694709946, 7.83870858996926130227690509142
