| L(s) = 1 | − i·2-s − 4-s + (−2.27 + 1.35i)7-s + i·8-s − 2.71i·11-s + 6.54i·13-s + (1.35 + 2.27i)14-s + 16-s + 1.53·17-s − 2.30i·19-s − 2.71·22-s − 3.83i·23-s + 6.54·26-s + (2.27 − 1.35i)28-s + 3.83i·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.858 + 0.512i)7-s + 0.353i·8-s − 0.817i·11-s + 1.81i·13-s + (0.362 + 0.607i)14-s + 0.250·16-s + 0.371·17-s − 0.528i·19-s − 0.577·22-s − 0.799i·23-s + 1.28·26-s + (0.429 − 0.256i)28-s + 0.711i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3724187562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3724187562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.27 - 1.35i)T \) |
| good | 11 | \( 1 + 2.71iT - 11T^{2} \) |
| 13 | \( 1 - 6.54iT - 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 19 | \( 1 + 2.30iT - 19T^{2} \) |
| 23 | \( 1 + 3.83iT - 23T^{2} \) |
| 29 | \( 1 - 3.83iT - 29T^{2} \) |
| 31 | \( 1 + 3.25iT - 31T^{2} \) |
| 37 | \( 1 + 3.01T + 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 - 0.468T + 43T^{2} \) |
| 47 | \( 1 + 9.11T + 47T^{2} \) |
| 53 | \( 1 + 9.25iT - 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 4.78iT - 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 2.30iT - 71T^{2} \) |
| 73 | \( 1 + 11.4iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−8.695711498302744718587060140263, −7.60401393857768542571409389944, −6.60043813849825049859810929751, −6.14401248608107669209125088720, −5.09125272803090781062993700264, −4.25414181236642658540131887046, −3.38254574773465320517974822313, −2.61474431480736526185906587950, −1.59462041276080583533582226366, −0.12268417269414241463177502929,
1.22459939472148983396711582909, 2.84676147510146393778375151789, 3.57723800880555471974376389807, 4.50247955257975115343642448363, 5.45486641967456931376781651165, 6.01994883956753411834475678579, 6.86955984552460581316101385567, 7.69252008446084970879427694372, 7.942806540994581625542625114215, 9.092872994464167326789012285720
