| L(s) = 1 | + (1.83 + 0.784i)2-s + (2.76 + 2.88i)4-s + 6.85·5-s − 3.48i·7-s + (2.82 + 7.48i)8-s + (12.6 + 5.37i)10-s + 1.50i·11-s + 0.937·13-s + (2.73 − 6.40i)14-s + (−0.666 + 15.9i)16-s + 17.5·17-s − 4.35i·19-s + (18.9 + 19.7i)20-s + (−1.18 + 2.76i)22-s + 9.27i·23-s + ⋯ |
| L(s) = 1 | + (0.919 + 0.392i)2-s + (0.692 + 0.721i)4-s + 1.37·5-s − 0.497i·7-s + (0.353 + 0.935i)8-s + (1.26 + 0.537i)10-s + 0.136i·11-s + 0.0721·13-s + (0.195 − 0.457i)14-s + (−0.0416 + 0.999i)16-s + 1.03·17-s − 0.229i·19-s + (0.949 + 0.989i)20-s + (−0.0536 + 0.125i)22-s + 0.403i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.257535793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.257535793\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.83 - 0.784i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - 6.85T + 25T^{2} \) |
| 7 | \( 1 + 3.48iT - 49T^{2} \) |
| 11 | \( 1 - 1.50iT - 121T^{2} \) |
| 13 | \( 1 - 0.937T + 169T^{2} \) |
| 17 | \( 1 - 17.5T + 289T^{2} \) |
| 23 | \( 1 - 9.27iT - 529T^{2} \) |
| 29 | \( 1 - 4.00T + 841T^{2} \) |
| 31 | \( 1 + 21.0iT - 961T^{2} \) |
| 37 | \( 1 + 13.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 74.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 13.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 12.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 48.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 20.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 53.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 136.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 70.1T + 9.40e3T^{2} \) |
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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
−10.32296200170985324272978142402, −9.701753887467573955681134222608, −8.535187264115404273658645930289, −7.50061904401824161818617778377, −6.66182940864445419716005594956, −5.77985248990155191505953234271, −5.13238023693426549565899750516, −3.92660143544946143724279288846, −2.76403009769737074210092980858, −1.56824195663550550773485143890,
1.36036082270667920671710120551, 2.39509524604027912669075015749, 3.41280944909639407248641774802, 4.82566762824663334447020597510, 5.66294239909892883235900493445, 6.18992194545025875887039135829, 7.25280340485443633178660605421, 8.627333715619884116180254081405, 9.621043977087976408955979617449, 10.22285741728347135510970629708
