| L(s) = 1 | + 25.5i·3-s − 131. i·7-s − 408.·9-s − 290.·11-s − 68.3i·13-s − 310. i·17-s − 2.13e3·19-s + 3.34e3·21-s + 873. i·23-s − 4.22e3i·27-s + 2.58e3·29-s + 9.08e3·31-s − 7.40e3i·33-s + 3.99e3i·37-s + 1.74e3·39-s + ⋯ |
| L(s) = 1 | + 1.63i·3-s − 1.01i·7-s − 1.68·9-s − 0.722·11-s − 0.112i·13-s − 0.260i·17-s − 1.35·19-s + 1.65·21-s + 0.344i·23-s − 1.11i·27-s + 0.569·29-s + 1.69·31-s − 1.18i·33-s + 0.479i·37-s + 0.183·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.575338608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575338608\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 25.5iT - 243T^{2} \) |
| 7 | \( 1 + 131. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 290.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 68.3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 310. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.13e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 873. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.99e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.69e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.80e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.48e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 7.65e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 9.23e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.32e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.23e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.65e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 7.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.96e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.18e4iT - 8.58e9T^{2} \) |
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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
−10.43800514640333166342319378974, −9.911958361849102785061731565543, −8.837849012162969235691991560916, −7.961171716266728517533417901333, −6.68724900715649043945795811498, −5.41521683724499492447045125230, −4.49365915576149627168737549827, −3.81503464938775282929079023862, −2.59424652335219409368416896673, −0.52816453530254608061078491357,
0.800203244962551492477048188173, 2.12253926755008464125400462307, 2.75679465594683325563082940666, 4.67802506741210012098525933425, 6.06250027595871168488203924663, 6.42621044521696871223912407085, 7.75377392283534741247603510379, 8.257041992476506873152136072098, 9.213208329148358968546102694561, 10.58407601812603119499970032844
