| L(s) = 1 | − 2-s + 4-s + (0.613 + 2.15i)5-s − 8-s + (−0.613 − 2.15i)10-s − 2.77i·11-s + 4.99·13-s + 16-s − 4.36i·17-s − 1.15i·19-s + (0.613 + 2.15i)20-s + 2.77i·22-s − 2.40·23-s + (−4.24 + 2.63i)25-s − 4.99·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.274 + 0.961i)5-s − 0.353·8-s + (−0.193 − 0.680i)10-s − 0.837i·11-s + 1.38·13-s + 0.250·16-s − 1.05i·17-s − 0.264i·19-s + (0.137 + 0.480i)20-s + 0.592i·22-s − 0.501·23-s + (−0.849 + 0.527i)25-s − 0.978·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8228632770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8228632770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.613 - 2.15i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.77iT - 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 + 4.36iT - 17T^{2} \) |
| 19 | \( 1 + 1.15iT - 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 + 6.90iT - 29T^{2} \) |
| 31 | \( 1 + 5.46iT - 31T^{2} \) |
| 37 | \( 1 - 0.263iT - 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 43 | \( 1 - 2.17iT - 43T^{2} \) |
| 47 | \( 1 + 6.93iT - 47T^{2} \) |
| 53 | \( 1 + 5.34T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 8.51iT - 61T^{2} \) |
| 67 | \( 1 - 11.9iT - 67T^{2} \) |
| 71 | \( 1 + 4.98iT - 71T^{2} \) |
| 73 | \( 1 + 9.07T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 9.01iT - 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 1.18T + 97T^{2} \) |
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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
−8.171459378212722259550556563894, −7.50960282898245036534562410100, −6.67578001476034857886890461358, −6.12485310525013900933202987610, −5.52975129988176238167665450024, −4.19306826953854155506415883416, −3.30963836364624860359786023724, −2.63640582086815174522286428881, −1.57146434466545605031792630555, −0.28680844751252415738458690520,
1.40557569910269972047175392095, 1.67068461350063304104348586004, 3.13743569750597093350908470829, 4.05689849605337364241711012538, 4.87072848275065618904575598700, 5.81912728612246291796411541010, 6.33319056421762396842202301714, 7.25768596330208839712404366579, 8.048669499056603090167815941899, 8.710428362835217632327365239856
