| L(s) = 1 | − i·2-s + (1 + i)3-s + 4-s + (1 − i)6-s + (3 − 3i)7-s − 3i·8-s − i·9-s + (−3 + 3i)11-s + (1 + i)12-s + (−3 − 3i)14-s − 16-s + (−4 − i)17-s − 18-s + 6i·19-s + 6·21-s + (3 + 3i)22-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.577 + 0.577i)3-s + 0.5·4-s + (0.408 − 0.408i)6-s + (1.13 − 1.13i)7-s − 1.06i·8-s − 0.333i·9-s + (−0.904 + 0.904i)11-s + (0.288 + 0.288i)12-s + (−0.801 − 0.801i)14-s − 0.250·16-s + (−0.970 − 0.242i)17-s − 0.235·18-s + 1.37i·19-s + 1.30·21-s + (0.639 + 0.639i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82680 - 0.891350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82680 - 0.891350i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + (4 + i)T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 3 | \( 1 + (-1 - i)T + 3iT^{2} \) |
| 7 | \( 1 + (-3 + 3i)T - 7iT^{2} \) |
| 11 | \( 1 + (3 - 3i)T - 11iT^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + (-1 + i)T - 23iT^{2} \) |
| 29 | \( 1 + (-3 - 3i)T + 29iT^{2} \) |
| 31 | \( 1 + (1 + i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 + (3 - 3i)T - 41iT^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + (-1 + i)T - 61iT^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + (-3 - 3i)T + 71iT^{2} \) |
| 73 | \( 1 + (3 + 3i)T + 73iT^{2} \) |
| 79 | \( 1 + (-7 + 7i)T - 79iT^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−10.88645862620098036861847931077, −10.26654997819384697253324179092, −9.615468957415970625998706701394, −8.270817963199271735754877227321, −7.50388973240751665204042357925, −6.49956799124622307315373326011, −4.76534627859259058160597149231, −4.02133466765917442898045466286, −2.80570019809810177639246574631, −1.48583317434370583270727717089,
2.07086679002784255870595377740, 2.71197284432485808348611293621, 4.93285456194737219430732325573, 5.61144542850559222660657547199, 6.78889168013275970491057941218, 7.71397706421828913638312442505, 8.434368629218127139620358629963, 8.873514114358976013800230556552, 10.77482184493561659314772457450, 11.19893813434124491607922070241
