L(s) = 1 | − 2.62i·3-s + (−2.19 − 0.432i)5-s − 0.864i·7-s − 3.89·9-s − 2·11-s + 2.62i·13-s + (−1.13 + 5.76i)15-s + i·17-s − 0.896·19-s − 2.27·21-s − 3.13i·23-s + (4.62 + 1.89i)25-s + 2.35i·27-s − 9.49·29-s − 9.01·31-s + ⋯ |
L(s) = 1 | − 1.51i·3-s + (−0.981 − 0.193i)5-s − 0.326i·7-s − 1.29·9-s − 0.603·11-s + 0.728i·13-s + (−0.293 + 1.48i)15-s + 0.242i·17-s − 0.205·19-s − 0.495·21-s − 0.653i·23-s + (0.925 + 0.379i)25-s + 0.453i·27-s − 1.76·29-s − 1.61·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1180443270\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1180443270\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.19 + 0.432i)T \) |
| 17 | \( 1 - iT \) |
good | 3 | \( 1 + 2.62iT - 3T^{2} \) |
| 7 | \( 1 + 0.864iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 19 | \( 1 + 0.896T + 19T^{2} \) |
| 23 | \( 1 + 3.13iT - 23T^{2} \) |
| 29 | \( 1 + 9.49T + 29T^{2} \) |
| 31 | \( 1 + 9.01T + 31T^{2} \) |
| 37 | \( 1 - 10.1iT - 37T^{2} \) |
| 41 | \( 1 - 9.52T + 41T^{2} \) |
| 43 | \( 1 - 7.25iT - 43T^{2} \) |
| 47 | \( 1 - 10.4iT - 47T^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 - 3.28T + 61T^{2} \) |
| 67 | \( 1 + 1.25iT - 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 2.62iT - 73T^{2} \) |
| 79 | \( 1 + 7.91T + 79T^{2} \) |
| 83 | \( 1 + 4.20iT - 83T^{2} \) |
| 89 | \( 1 + 4.14T + 89T^{2} \) |
| 97 | \( 1 + 6.14iT - 97T^{2} \) |
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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
−9.624067728601352863897896476497, −8.674814356846770301827755717307, −7.915151910951105605725888959214, −7.42355990146483347136828295213, −6.74388195949769327920756389328, −5.84673806518055533830761379315, −4.66897818749949371004073317590, −3.67210476977398557031438283660, −2.42871394457904191020765354441, −1.30199162640785003686256707054,
0.05019842757079758475418208918, 2.49337246462762306193163454598, 3.66618782113900444723781412532, 3.99840276983545780694847832109, 5.30584771369593078089575403344, 5.59820906850569162716983756713, 7.25959060540142362253374285003, 7.77600772746028464001186123328, 8.939339707908594735355960071645, 9.279429168984608026781937040059