| L(s) = 1 | + 5.99·5-s + (−15.8 + 9.58i)7-s − 45.4i·11-s − 26.2i·13-s + 19.7·17-s − 27.9i·19-s + 69.6i·23-s − 89.0·25-s − 137. i·29-s + 159. i·31-s + (−95.0 + 57.4i)35-s − 287.·37-s − 40.8·41-s − 110.·43-s + 219.·47-s + ⋯ |
| L(s) = 1 | + 0.536·5-s + (−0.855 + 0.517i)7-s − 1.24i·11-s − 0.560i·13-s + 0.281·17-s − 0.337i·19-s + 0.631i·23-s − 0.712·25-s − 0.880i·29-s + 0.924i·31-s + (−0.459 + 0.277i)35-s − 1.27·37-s − 0.155·41-s − 0.393·43-s + 0.680·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6765074808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6765074808\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (15.8 - 9.58i)T \) |
| good | 5 | \( 1 - 5.99T + 125T^{2} \) |
| 11 | \( 1 + 45.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 26.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 19.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 69.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 137. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 159. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 287.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 40.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 110.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 219.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 209. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 827.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 686. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 342.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 387. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 220. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 484.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 708.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 140.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.50e3iT - 9.12e5T^{2} \) |
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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
−8.802761334852390224447865072346, −8.415328184595873619461971403333, −7.30637512911782572078745919283, −6.49393009688924280095545714777, −5.67440749780298836371684676259, −5.36063453189970203548486525474, −3.85464246531922037204135853361, −3.15028948480783971182691799381, −2.28415640114070338534169103945, −0.973015013487545249409156853432,
0.14428742426044034454381700820, 1.54074126029764269534479145367, 2.38098946867454292700160232076, 3.55254449514277792876992023310, 4.31123833147907583697028258949, 5.25797088680682793702544564652, 6.16732433749181512831539684677, 6.90012429342589055010038342381, 7.42755275824136820845286009141, 8.485729494980236035825721284857
