L(s) = 1 | − i·3-s + i·5-s − 0.941·7-s − 9-s − 4.49i·11-s − 5.55i·13-s + 15-s + 7.55·17-s − 1.05i·19-s + 0.941i·21-s + 1.05·23-s − 25-s + i·27-s − 2i·29-s − 3.55·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.447i·5-s − 0.355·7-s − 0.333·9-s − 1.35i·11-s − 1.54i·13-s + 0.258·15-s + 1.83·17-s − 0.242i·19-s + 0.205i·21-s + 0.220·23-s − 0.200·25-s + 0.192i·27-s − 0.371i·29-s − 0.638·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.983871 - 0.814131i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.983871 - 0.814131i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 7 | \( 1 + 0.941T + 7T^{2} \) |
| 11 | \( 1 + 4.49iT - 11T^{2} \) |
| 13 | \( 1 + 5.55iT - 13T^{2} \) |
| 17 | \( 1 - 7.55T + 17T^{2} \) |
| 19 | \( 1 + 1.05iT - 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 3.55T + 31T^{2} \) |
| 37 | \( 1 + 7.43iT - 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 - 1.88iT - 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 8.49iT - 59T^{2} \) |
| 61 | \( 1 - 8.99iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 5.88iT - 83T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−10.75780938904668132629841577795, −10.08773788394200858082772270001, −8.878736721759559883120908697887, −7.937502629637777766523348748891, −7.27904237937746686424704423113, −5.93660377350847510819710304981, −5.54172097898345409358064700975, −3.54699099817499929630803780065, −2.82786291525377585220673924844, −0.827127628475112155690406004850,
1.75494050146072302743925958027, 3.43352141847321001289160477458, 4.48620098916828239131510347841, 5.34131386413595114603393540565, 6.60291025184386348034570315215, 7.53873771101087929450734618509, 8.662880444990517793904317842654, 9.690565645499377432218349214238, 9.912464874436402132391434452943, 11.20958581452967761448997558502