L(s) = 1 | + i·3-s + (−1.48 − 1.66i)5-s + 2.27i·7-s − 9-s + 3.88·11-s + 2.72i·13-s + (1.66 − 1.48i)15-s + 3.85i·17-s + 1.23·19-s − 2.27·21-s + 0.161i·23-s + (−0.569 + 4.96i)25-s − i·27-s − 9.75·29-s + 10.8·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.665 − 0.746i)5-s + 0.861i·7-s − 0.333·9-s + 1.17·11-s + 0.754i·13-s + (0.430 − 0.384i)15-s + 0.934i·17-s + 0.283·19-s − 0.497·21-s + 0.0337i·23-s + (−0.113 + 0.993i)25-s − 0.192i·27-s − 1.81·29-s + 1.94·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.203996138\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.203996138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.48 + 1.66i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 - 2.27iT - 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 - 2.72iT - 13T^{2} \) |
| 17 | \( 1 - 3.85iT - 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 - 0.161iT - 23T^{2} \) |
| 29 | \( 1 + 9.75T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 4.23iT - 37T^{2} \) |
| 41 | \( 1 + 9.12T + 41T^{2} \) |
| 43 | \( 1 + 12.4iT - 43T^{2} \) |
| 47 | \( 1 - 7.41iT - 47T^{2} \) |
| 53 | \( 1 - 5.06iT - 53T^{2} \) |
| 59 | \( 1 + 0.0652T + 59T^{2} \) |
| 61 | \( 1 - 7.45T + 61T^{2} \) |
| 71 | \( 1 - 3.87T + 71T^{2} \) |
| 73 | \( 1 - 2.30iT - 73T^{2} \) |
| 79 | \( 1 + 8.08T + 79T^{2} \) |
| 83 | \( 1 - 8.86iT - 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 - 1.15iT - 97T^{2} \) |
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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
−8.824962856457366544340659141423, −8.233418730467765297119523533797, −7.29334603654189877035417200670, −6.41410817688193719721919298033, −5.65525448215943421119143408827, −4.92766528178670508426813487245, −3.98780285965123950603422689492, −3.69032418684176798993957434175, −2.30322014621549856075226410957, −1.25501429881619787428823335275,
0.38047454202608651937071546207, 1.41828966001710017464711566441, 2.76182575158533446119291744870, 3.46514958711627672669192199661, 4.22230990446416239217050876693, 5.16947936703301928993385218055, 6.27914722846150675493916460759, 6.80192481647247192086609396880, 7.38677611007091125099481890122, 7.996085865289634429353643489383