| L(s) = 1 | − i·2-s − 4-s + i·7-s + i·8-s − i·9-s + 11-s + 14-s + 16-s − 18-s − i·22-s + 2·23-s − i·25-s − i·28-s + (−1 + i)29-s − i·32-s + ⋯ |
| L(s) = 1 | − i·2-s − 4-s + i·7-s + i·8-s − i·9-s + 11-s + 14-s + 16-s − 18-s − i·22-s + 2·23-s − i·25-s − i·28-s + (−1 + i)29-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.019485545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019485545\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - 2T + T^{2} \) |
| 29 | \( 1 + (1 - i)T - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.534442675177185783150692025598, −9.109410981963929109274314474572, −8.661309859231122073728848479951, −7.33719140968256154201337727478, −6.27159051587282026080415053505, −5.45234931269963933248496686931, −4.38452923970129874263346732199, −3.45348967165494138560218333538, −2.54726031371916870808363126131, −1.20254169182848251670829448521,
1.30156690402240186779579934073, 3.24038722068773261380307460293, 4.30556974626068662195768103044, 4.94728203897349001685731397909, 6.01201692798405226797944268329, 6.91641561104544527958793937207, 7.50334964322855352550380437651, 8.214986974922448394767389485099, 9.267068946391408351457225898920, 9.745197704082420419338409077111
