| L(s) = 1 | + (2.84 + 2.84i)7-s − 3i·9-s − 6.50·11-s + (−1.69 − 1.69i)17-s + 4.35i·19-s + (−6.35 + 6.35i)23-s + (−8.74 + 8.74i)43-s + (−5.35 − 5.35i)47-s + 9.20i·49-s + 10.8·61-s + (8.54 − 8.54i)63-s + (−5.11 + 5.11i)73-s + (−18.5 − 18.5i)77-s − 9·81-s + (3.64 − 3.64i)83-s + ⋯ |
| L(s) = 1 | + (1.07 + 1.07i)7-s − i·9-s − 1.96·11-s + (−0.411 − 0.411i)17-s + 0.999i·19-s + (−1.32 + 1.32i)23-s + (−1.33 + 1.33i)43-s + (−0.781 − 0.781i)47-s + 1.31i·49-s + 1.38·61-s + (1.07 − 1.07i)63-s + (−0.599 + 0.599i)73-s + (−2.11 − 2.11i)77-s − 81-s + (0.399 − 0.399i)83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6196073979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6196073979\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 + 3iT^{2} \) |
| 7 | \( 1 + (-2.84 - 2.84i)T + 7iT^{2} \) |
| 11 | \( 1 + 6.50T + 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (1.69 + 1.69i)T + 17iT^{2} \) |
| 23 | \( 1 + (6.35 - 6.35i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (8.74 - 8.74i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.35 + 5.35i)T + 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (5.11 - 5.11i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-3.64 + 3.64i)T - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−9.653072544259291435374474061583, −8.501299869252857778669276475106, −8.143909447902112453737295823976, −7.36998849066128101201013160693, −6.15625555729494384801904896833, −5.48084594160656701613480402694, −4.88154046377100977520222689703, −3.65065347382710000485067047629, −2.60833677860204946512164595064, −1.68386492492102590525204831014,
0.20644170500683118691884435256, 1.88629862452995076166787757714, 2.67420878751827380721629556917, 4.13686296552621376223864588351, 4.84446534839111008941027002624, 5.38464498531375580360157642280, 6.66647108976640028774202296109, 7.53035736175766683586970954392, 8.093433498983748076063390590313, 8.516758159070589924168548890833
