L(s) = 1 | − 2.64·2-s + 4.97·4-s − 1.42·5-s + 2.00·7-s − 7.85·8-s + 3.75·10-s − 1.91·11-s + 5.67·13-s − 5.28·14-s + 10.8·16-s + 5.11·17-s − 0.0723·19-s − 7.07·20-s + 5.05·22-s − 23-s − 2.97·25-s − 14.9·26-s + 9.96·28-s + 29-s + 10.4·31-s − 12.8·32-s − 13.5·34-s − 2.84·35-s + 0.0359·37-s + 0.191·38-s + 11.1·40-s + 11.1·41-s + ⋯ |
L(s) = 1 | − 1.86·2-s + 2.48·4-s − 0.636·5-s + 0.756·7-s − 2.77·8-s + 1.18·10-s − 0.577·11-s + 1.57·13-s − 1.41·14-s + 2.70·16-s + 1.23·17-s − 0.0165·19-s − 1.58·20-s + 1.07·22-s − 0.208·23-s − 0.595·25-s − 2.94·26-s + 1.88·28-s + 0.185·29-s + 1.88·31-s − 2.26·32-s − 2.31·34-s − 0.481·35-s + 0.00590·37-s + 0.0309·38-s + 1.76·40-s + 1.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8960723121\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8960723121\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 5 | \( 1 + 1.42T + 5T^{2} \) |
| 7 | \( 1 - 2.00T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 - 5.67T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 19 | \( 1 + 0.0723T + 19T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 0.0359T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 8.46T + 47T^{2} \) |
| 53 | \( 1 - 9.11T + 53T^{2} \) |
| 59 | \( 1 - 6.17T + 59T^{2} \) |
| 61 | \( 1 - 5.91T + 61T^{2} \) |
| 67 | \( 1 + 1.78T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 5.99T + 73T^{2} \) |
| 79 | \( 1 - 7.00T + 79T^{2} \) |
| 83 | \( 1 + 7.42T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 1.32T + 97T^{2} \) |
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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
−8.242755472438142189109996234140, −7.73473559810941035380851374560, −7.04203225340052712586004325168, −6.15150252151998589582650579521, −5.56424237918238007438201717354, −4.31176815734485972514616306688, −3.36528644180980225061414138074, −2.46878057469903653969897895259, −1.40356847071586852115651081024, −0.73459089646811826158849287771,
0.73459089646811826158849287771, 1.40356847071586852115651081024, 2.46878057469903653969897895259, 3.36528644180980225061414138074, 4.31176815734485972514616306688, 5.56424237918238007438201717354, 6.15150252151998589582650579521, 7.04203225340052712586004325168, 7.73473559810941035380851374560, 8.242755472438142189109996234140