| L(s) = 1 | + i·2-s + i·3-s + 4-s − 6-s + i·7-s + 3i·8-s − 9-s − 11-s + i·12-s − i·13-s − 14-s − 16-s + 7i·17-s − i·18-s − 21-s − i·22-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s + 0.5·4-s − 0.408·6-s + 0.377i·7-s + 1.06i·8-s − 0.333·9-s − 0.301·11-s + 0.288i·12-s − 0.277i·13-s − 0.267·14-s − 0.250·16-s + 1.69i·17-s − 0.235i·18-s − 0.218·21-s − 0.213i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.375185 + 1.58930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375185 + 1.58930i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 5iT - 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 11iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−10.52434940516577168327865340994, −9.446759461177391178498549445805, −8.486204971751738744300218392421, −7.940697842689239057631078536823, −6.93422604418664686580821272631, −5.94396543644895071112042104231, −5.48291234248810505084750319914, −4.27103674759231294887935414309, −3.10336653665847457017880378324, −1.91696029519840575915877632473,
0.72386012556167078045287354427, 2.06797748971956842010687938389, 2.94085915972937908553601403264, 4.05005750577936122211108810791, 5.31681160649411163537451040130, 6.36917371395779687458389034173, 7.27344276489352481846552249237, 7.65759326503720371265935522019, 9.065418090989400501468343663135, 9.701763933474801837759326348018
