L(s) = 1 | + (0.707 + 0.707i)2-s + (2.70 − 1.29i)3-s − 3i·4-s + (0.707 + 4.94i)5-s + (2.82 + i)6-s + 7i·7-s + (4.94 − 4.94i)8-s + (5.65 − 7i)9-s + (−3.00 + 4.00i)10-s − 9.89i·11-s + (−3.87 − 8.12i)12-s + (8 − 8i)13-s + (−4.94 + 4.94i)14-s + (8.31 + 12.4i)15-s − 4.99·16-s + (−18.3 + 18.3i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.353i)2-s + (0.902 − 0.430i)3-s − 0.750i·4-s + (0.141 + 0.989i)5-s + (0.471 + 0.166i)6-s + i·7-s + (0.618 − 0.618i)8-s + (0.628 − 0.777i)9-s + (−0.300 + 0.400i)10-s − 0.899i·11-s + (−0.323 − 0.676i)12-s + (0.615 − 0.615i)13-s + (−0.353 + 0.353i)14-s + (0.554 + 0.832i)15-s − 0.312·16-s + (−1.08 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.09602 + 0.0355141i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.09602 + 0.0355141i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.70 + 1.29i)T \) |
| 5 | \( 1 + (-0.707 - 4.94i)T \) |
| 7 | \( 1 - 7iT \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T + 4iT^{2} \) |
| 11 | \( 1 + 9.89iT - 121T^{2} \) |
| 13 | \( 1 + (-8 + 8i)T - 169iT^{2} \) |
| 17 | \( 1 + (18.3 - 18.3i)T - 289iT^{2} \) |
| 19 | \( 1 + 10T + 361T^{2} \) |
| 23 | \( 1 + (24.0 - 24.0i)T - 529iT^{2} \) |
| 29 | \( 1 + 4.24T + 841T^{2} \) |
| 31 | \( 1 - 14iT - 961T^{2} \) |
| 37 | \( 1 + (-30 + 30i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 33.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36 - 36i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (4.24 - 4.24i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-70.7 + 70.7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 29.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 14iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-32 + 32i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 59.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-39 + 39i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 56iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (36.7 + 36.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 19.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-113 - 113i)T + 9.40e3iT^{2} \) |
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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
−13.69018909164041274046321438987, −12.94314844795436462932885353596, −11.33982009597783489700585662681, −10.31566467208221908516233569754, −9.089146681742194004043493140220, −7.990704370693596228110361884251, −6.48996326215910730911536666978, −5.85132540180852832653601515936, −3.68280546313233073517983183157, −2.08613333671172545868103919682,
2.19475981152316429990697946482, 4.11642442026930395648732055817, 4.54679444261208009815435983861, 7.05945642683123586463011996937, 8.193493020369552392437613479497, 9.091765553966118946111961630164, 10.25481035052870926872481931563, 11.55104778079896122386114604078, 12.75942651533726075640658333542, 13.48925061161468557548554797153