| L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 3.12i·5-s + 18.5·7-s + 22.6i·8-s − 8.83·10-s + 141. i·11-s − 237.·13-s − 52.3i·14-s + 64.0·16-s − 394. i·17-s + 535.·19-s + 24.9i·20-s + 399.·22-s − 487. i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.124i·5-s + 0.377·7-s + 0.353i·8-s − 0.0883·10-s + 1.16i·11-s − 1.40·13-s − 0.267i·14-s + 0.250·16-s − 1.36i·17-s + 1.48·19-s + 0.0624i·20-s + 0.825·22-s − 0.922i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.735166221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.735166221\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 18.5T \) |
| good | 5 | \( 1 + 3.12iT - 625T^{2} \) |
| 11 | \( 1 - 141. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 237.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 394. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 535.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 487. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.20e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 296.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 380.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 73.6iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.91e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.84e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.12e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.48e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.03e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.34e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 8.60e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.16e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.79e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 2.41e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.37e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.06e4T + 8.85e7T^{2} \) |
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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
−10.44942075322533712435678843689, −9.673289289774401291761896807214, −8.977654989018313465939235110399, −7.61056376847826960829489339485, −6.97443845668838276998404901059, −5.04642681252211928308151139578, −4.79658657927512058420665717745, −3.11699167478030340110464905887, −2.07371700635607461067188461107, −0.62457226908631819786229960391,
1.00285219598287761496904386527, 2.81506357174174296847056613641, 4.13228401241459460325864897792, 5.33468663190820876002492855372, 6.08839829813382690180738917483, 7.36095595098838234184190699548, 7.976761763694469571762801094497, 9.027041867575505351787770201805, 9.912252040574179845383198847193, 10.93744958514628521449300464950
