| L(s) = 1 | + 0.656i·2-s + 1.56·4-s − 3.31·5-s + 2.34i·8-s − 2.17i·10-s − 2.34i·11-s − 1.58i·13-s + 1.59·16-s − 1.13·17-s + 4.44i·19-s − 5.19·20-s + 1.53·22-s + 9.39i·23-s + 5.96·25-s + 1.03·26-s + ⋯ |
| L(s) = 1 | + 0.464i·2-s + 0.784·4-s − 1.48·5-s + 0.828i·8-s − 0.687i·10-s − 0.706i·11-s − 0.438i·13-s + 0.399·16-s − 0.275·17-s + 1.02i·19-s − 1.16·20-s + 0.328·22-s + 1.95i·23-s + 1.19·25-s + 0.203·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9818601372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9818601372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.656iT - 2T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 11 | \( 1 + 2.34iT - 11T^{2} \) |
| 13 | \( 1 + 1.58iT - 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 - 4.44iT - 19T^{2} \) |
| 23 | \( 1 - 9.39iT - 23T^{2} \) |
| 29 | \( 1 - 3.65iT - 29T^{2} \) |
| 31 | \( 1 - 7.31iT - 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 - 4.00iT - 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 6.39iT - 71T^{2} \) |
| 73 | \( 1 - 10.6iT - 73T^{2} \) |
| 79 | \( 1 - 1.23T + 79T^{2} \) |
| 83 | \( 1 - 1.03T + 83T^{2} \) |
| 89 | \( 1 + 7.47T + 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−10.12299960425030032299033165487, −8.693018426664508463615789333172, −8.244334197646868019814341623508, −7.41854110084623866485736797474, −6.93807175192706454557102796957, −5.80718291117837122955898143063, −5.07264259491951435189798645471, −3.64130405542562105693133222367, −3.20824110361942394457411843417, −1.51972642147644876863059866097,
0.39526755961886483477677977041, 2.07158984630470129149480897224, 3.03833820641183611987065308359, 4.12169773223595724500799369931, 4.68562268388500620132828367289, 6.26317773311635281247796044925, 6.98265334262250659968143502212, 7.58845528913158108446961071884, 8.445104658520951505966527438143, 9.347354635597486351026468958116
