L(s) = 1 | + 6.84·5-s + 3.94·7-s − 15.1·11-s − 1.55i·13-s − 2.94·17-s + (−16.0 − 10.1i)19-s − 20.2·23-s + 21.9·25-s − 4.45i·29-s + 3.63i·31-s + 27.0·35-s + 17.6i·37-s + 76.0i·41-s − 26.7·43-s − 48.9·47-s + ⋯ |
L(s) = 1 | + 1.36·5-s + 0.564·7-s − 1.37·11-s − 0.119i·13-s − 0.173·17-s + (−0.846 − 0.533i)19-s − 0.879·23-s + 0.876·25-s − 0.153i·29-s + 0.117i·31-s + 0.772·35-s + 0.477i·37-s + 1.85i·41-s − 0.621·43-s − 1.04·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.533i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6790539842\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6790539842\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (16.0 + 10.1i)T \) |
good | 5 | \( 1 - 6.84T + 25T^{2} \) |
| 7 | \( 1 - 3.94T + 49T^{2} \) |
| 11 | \( 1 + 15.1T + 121T^{2} \) |
| 13 | \( 1 + 1.55iT - 169T^{2} \) |
| 17 | \( 1 + 2.94T + 289T^{2} \) |
| 23 | \( 1 + 20.2T + 529T^{2} \) |
| 29 | \( 1 + 4.45iT - 841T^{2} \) |
| 31 | \( 1 - 3.63iT - 961T^{2} \) |
| 37 | \( 1 - 17.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 76.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 26.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 48.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 49.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 97.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 105.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 129. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 130. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 15.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 62.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 134. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 116. iT - 9.40e3T^{2} \) |
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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
−8.947994191487107361280037790398, −8.198495367222973539105252230751, −7.57912335006764524933339972295, −6.43418167383099446238711013916, −5.95976293889117075217121554016, −5.04643637774681456451789234561, −4.53625407783715466935285814138, −3.03179668699804211614165814330, −2.28359577113474650741824420035, −1.45635668620838897693280777254,
0.13185267270563884268271335206, 1.83964209562744990597472380666, 2.15710360823120919971395382399, 3.38380599347498942512853825756, 4.60149092094940845080247779363, 5.30551281031366038741720126786, 5.94108203723576489398689483217, 6.67254740141008191130260119559, 7.73326292051563864091114867998, 8.289261721196410761688013291910