L(s) = 1 | − 2-s + 4-s + (1.35 + 2.27i)7-s − 8-s − 2.71i·11-s − 6.54·13-s + (−1.35 − 2.27i)14-s + 16-s − 1.53i·17-s + 2.30i·19-s + 2.71i·22-s + 3.83·23-s + 6.54·26-s + (1.35 + 2.27i)28-s − 3.83i·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.512 + 0.858i)7-s − 0.353·8-s − 0.817i·11-s − 1.81·13-s + (−0.362 − 0.607i)14-s + 0.250·16-s − 0.371i·17-s + 0.528i·19-s + 0.577i·22-s + 0.799·23-s + 1.28·26-s + (0.256 + 0.429i)28-s − 0.711i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.083757459\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.083757459\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-1.35 - 2.27i)T \) |
good | 11 | \( 1 + 2.71iT - 11T^{2} \) |
| 13 | \( 1 + 6.54T + 13T^{2} \) |
| 17 | \( 1 + 1.53iT - 17T^{2} \) |
| 19 | \( 1 - 2.30iT - 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 + 3.83iT - 29T^{2} \) |
| 31 | \( 1 + 3.25iT - 31T^{2} \) |
| 37 | \( 1 - 3.01iT - 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 - 0.468iT - 43T^{2} \) |
| 47 | \( 1 - 9.11iT - 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 4.78iT - 61T^{2} \) |
| 67 | \( 1 - 13.5iT - 67T^{2} \) |
| 71 | \( 1 + 2.30iT - 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.0iT - 83T^{2} \) |
| 89 | \( 1 + 9.60T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.791420311585288508621089838897, −8.093183913630325778089405548818, −7.51022083238716365684653721235, −6.69226873862180267224579888998, −5.71603938509943319366083065313, −5.18420222885300563499636355807, −4.14508462237286683776788331832, −2.76710447417815955208609421116, −2.34352759404373788869857308782, −0.931113808251620411651497309231,
0.52175841863948880570758235392, 1.79995089751458059554665022770, 2.63200462880158691466371484138, 3.84644677141544734854400854225, 4.80985471515983507946398251857, 5.33012787480226294759947329895, 6.82349633059484719405233566350, 7.07941293468911438381637262625, 7.73343337626313419319954814003, 8.541188132472248471704927571372