| L(s) = 1 | − 1.41i·5-s + 10·13-s − 9.89i·17-s + 23·25-s − 1.41i·29-s − 24·37-s + 43.8i·41-s − 49·49-s − 103. i·53-s + 120·61-s − 14.1i·65-s + 96·73-s − 14.0·85-s + 57.9i·89-s − 144·97-s + ⋯ |
| L(s) = 1 | − 0.282i·5-s + 0.769·13-s − 0.582i·17-s + 0.920·25-s − 0.0487i·29-s − 0.648·37-s + 1.06i·41-s − 0.999·49-s − 1.94i·53-s + 1.96·61-s − 0.217i·65-s + 1.31·73-s − 0.164·85-s + 0.651i·89-s − 1.48·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.979859741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979859741\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41iT - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 + 9.89iT - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 1.41iT - 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 + 24T + 1.36e3T^{2} \) |
| 41 | \( 1 - 43.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 103. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 120T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 57.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 144T + 9.40e3T^{2} \) |
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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
−8.568708579628585631455101596033, −8.160307726540877938062077772303, −7.04760156382425246952166526087, −6.48358926373150078964595586580, −5.45792172537949044686615948224, −4.79772981479826657594582065140, −3.78840346633234892473676444728, −2.91738824568286215836848926407, −1.70719786381970485182183148789, −0.58609097670388090670597513836,
0.966529855225046453087715220503, 2.12276037148964981375399013133, 3.23117559141155859592590978611, 3.98934704598245933147996809130, 5.00608862267483122945426146925, 5.87907337304749731098985815398, 6.61893045178887135210063251452, 7.34560988493813313265906340013, 8.291171142127193147711532692932, 8.829474484440899517734602056623
