L(s) = 1 | − 4·7-s − 2i·13-s + 8i·19-s + 5·25-s + 4·31-s − 10i·37-s − 8i·43-s + 9·49-s − 14i·61-s − 16i·67-s + 10·73-s + 4·79-s + 8i·91-s + 14·97-s + 20·103-s + ⋯ |
L(s) = 1 | − 1.51·7-s − 0.554i·13-s + 1.83i·19-s + 25-s + 0.718·31-s − 1.64i·37-s − 1.21i·43-s + 1.28·49-s − 1.79i·61-s − 1.95i·67-s + 1.17·73-s + 0.450·79-s + 0.838i·91-s + 1.42·97-s + 1.97·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.214325323\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.214325323\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 14iT - 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.008428359699540435016744051413, −8.124169989877371713769418839823, −7.35928195643356245391222192985, −6.44909564911114125700282328747, −5.94757960444598584508574878041, −5.02021312284690983688973493731, −3.74383113409750442137554660928, −3.29021543040717627143857293621, −2.11261587435062822607495516605, −0.54054744680117114954700093718,
0.907548119678773926813708962442, 2.59733218933344662632698734592, 3.14661758904341400413257763534, 4.29872888615737020177104784318, 5.06280644591501156631301521343, 6.26610341501198060980853030440, 6.66926198963039532733097882902, 7.35968717154217309943927251970, 8.554947232483548683002580658808, 9.109355025347877427749698838547