| L(s) = 1 | − 0.898·3-s − 5·5-s − 28.4·7-s − 26.1·9-s + 2.60·11-s + 33.1·13-s + 4.49·15-s + 119.·17-s + 143.·19-s + 25.6·21-s − 113.·23-s + 25·25-s + 47.8·27-s + 67.6·29-s − 117.·31-s − 2.34·33-s + 142.·35-s + 256.·37-s − 29.8·39-s + 245.·41-s − 369.·43-s + 130.·45-s + 76.3·47-s + 468.·49-s − 107.·51-s − 40.4·53-s − 13.0·55-s + ⋯ |
| L(s) = 1 | − 0.173·3-s − 0.447·5-s − 1.53·7-s − 0.970·9-s + 0.0714·11-s + 0.708·13-s + 0.0773·15-s + 1.70·17-s + 1.72·19-s + 0.266·21-s − 1.03·23-s + 0.200·25-s + 0.340·27-s + 0.432·29-s − 0.680·31-s − 0.0123·33-s + 0.688·35-s + 1.13·37-s − 0.122·39-s + 0.936·41-s − 1.30·43-s + 0.433·45-s + 0.236·47-s + 1.36·49-s − 0.295·51-s − 0.104·53-s − 0.0319·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.160940264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160940264\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 3 | \( 1 + 0.898T + 27T^{2} \) |
| 7 | \( 1 + 28.4T + 343T^{2} \) |
| 11 | \( 1 - 2.60T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 67.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 256.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 245.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 76.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 40.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 457.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 477.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 602.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 117.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 858.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 565.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 625.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 805.T + 9.12e5T^{2} \) |
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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
−11.39242692884666210346003038097, −10.10735698523363330372162515072, −9.501562644064650117167988127424, −8.335629326804172738335076462841, −7.36634832849761104567436542577, −6.16405594571565973389345292912, −5.48418352571995065084059071778, −3.67535247836420756118270682923, −3.01298621773589019788955924139, −0.73667499193239584424072505830,
0.73667499193239584424072505830, 3.01298621773589019788955924139, 3.67535247836420756118270682923, 5.48418352571995065084059071778, 6.16405594571565973389345292912, 7.36634832849761104567436542577, 8.335629326804172738335076462841, 9.501562644064650117167988127424, 10.10735698523363330372162515072, 11.39242692884666210346003038097
