| L(s) = 1 | + 7·7-s + 54·11-s + 55·13-s + 18·17-s − 25·19-s − 18·23-s + 54·29-s − 271·31-s − 314·37-s + 360·41-s + 163·43-s + 522·47-s − 294·49-s − 36·53-s − 126·59-s + 47·61-s + 343·67-s + 1.08e3·71-s + 1.05e3·73-s + 378·77-s − 568·79-s + 1.42e3·83-s − 1.44e3·89-s + 385·91-s + 439·97-s − 828·101-s − 548·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.48·11-s + 1.17·13-s + 0.256·17-s − 0.301·19-s − 0.163·23-s + 0.345·29-s − 1.57·31-s − 1.39·37-s + 1.37·41-s + 0.578·43-s + 1.62·47-s − 6/7·49-s − 0.0933·53-s − 0.278·59-s + 0.0986·61-s + 0.625·67-s + 1.80·71-s + 1.68·73-s + 0.559·77-s − 0.808·79-s + 1.88·83-s − 1.71·89-s + 0.443·91-s + 0.459·97-s − 0.815·101-s − 0.524·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.573318861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.573318861\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 - 54 T + p^{3} T^{2} \) |
| 13 | \( 1 - 55 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 25 T + p^{3} T^{2} \) |
| 23 | \( 1 + 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 271 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 360 T + p^{3} T^{2} \) |
| 43 | \( 1 - 163 T + p^{3} T^{2} \) |
| 47 | \( 1 - 522 T + p^{3} T^{2} \) |
| 53 | \( 1 + 36 T + p^{3} T^{2} \) |
| 59 | \( 1 + 126 T + p^{3} T^{2} \) |
| 61 | \( 1 - 47 T + p^{3} T^{2} \) |
| 67 | \( 1 - 343 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1080 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1054 T + p^{3} T^{2} \) |
| 79 | \( 1 + 568 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1422 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1440 T + p^{3} T^{2} \) |
| 97 | \( 1 - 439 T + p^{3} T^{2} \) |
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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
−9.537168433030510841079230412424, −8.934424847847224782693482416561, −8.149592574864072632047449826793, −7.09642625345007426732263628219, −6.29288245584064244171628106449, −5.43496859595850097251987885510, −4.16502834221839074961476442356, −3.53207092300331186555514172272, −1.95118381423557312230889772872, −0.934833680763450745019194885713,
0.934833680763450745019194885713, 1.95118381423557312230889772872, 3.53207092300331186555514172272, 4.16502834221839074961476442356, 5.43496859595850097251987885510, 6.29288245584064244171628106449, 7.09642625345007426732263628219, 8.149592574864072632047449826793, 8.934424847847224782693482416561, 9.537168433030510841079230412424
