L(s) = 1 | − 0.327i·3-s + i·5-s − 2.33·7-s + 2.89·9-s − 3.77·11-s + 1.12·13-s + 0.327·15-s − 2.52i·17-s − 3.12·19-s + 0.765i·21-s + (1.12 + 4.66i)23-s − 25-s − 1.92i·27-s − 4.41·29-s − 2.01i·31-s + ⋯ |
L(s) = 1 | − 0.188i·3-s + 0.447i·5-s − 0.884·7-s + 0.964·9-s − 1.13·11-s + 0.312·13-s + 0.0844·15-s − 0.612i·17-s − 0.716·19-s + 0.166i·21-s + (0.235 + 0.971i)23-s − 0.200·25-s − 0.370i·27-s − 0.820·29-s − 0.361i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1424126421\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1424126421\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-1.12 - 4.66i)T \) |
good | 3 | \( 1 + 0.327iT - 3T^{2} \) |
| 7 | \( 1 + 2.33T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 - 1.12T + 13T^{2} \) |
| 17 | \( 1 + 2.52iT - 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 + 2.01iT - 31T^{2} \) |
| 37 | \( 1 + 7.54iT - 37T^{2} \) |
| 41 | \( 1 + 4.88T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 - 7.22iT - 47T^{2} \) |
| 53 | \( 1 - 4.78iT - 53T^{2} \) |
| 59 | \( 1 + 9.02iT - 59T^{2} \) |
| 61 | \( 1 + 3.02iT - 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 + 9.56iT - 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 + 4.80T + 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 97 | \( 1 - 8.55iT - 97T^{2} \) |
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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
−9.041761492612662084565283962982, −7.84820286355651749075839183176, −7.35679892572260938613149909538, −6.56348825403173968709712121903, −5.78051880556055686068749960891, −4.79226673390897522862228882159, −3.72656369189406613643498120141, −2.89536436174986917112322127798, −1.78060051588760154718197622999, −0.04990858793596745109997633178,
1.57954760872686241893644666680, 2.83690782326319791560246034481, 3.85140051078674747744633728799, 4.67700328673469258390800961876, 5.54630157388173044466979918255, 6.51030379065167314010216918344, 7.14556084748813341663778458226, 8.233640352989418686629462007337, 8.716654476889363425586629013877, 9.806769913227057516593620178365