L(s) = 1 | + (1.70 + 0.292i)3-s − 0.585i·5-s − i·7-s + (2.82 + i)9-s + 2·11-s − 0.585·13-s + (0.171 − i)15-s − 0.828i·17-s + 2.24i·19-s + (0.292 − 1.70i)21-s + 4·23-s + 4.65·25-s + (4.53 + 2.53i)27-s + 0.828i·29-s − 6.82i·31-s + ⋯ |
L(s) = 1 | + (0.985 + 0.169i)3-s − 0.261i·5-s − 0.377i·7-s + (0.942 + 0.333i)9-s + 0.603·11-s − 0.162·13-s + (0.0442 − 0.258i)15-s − 0.200i·17-s + 0.514i·19-s + (0.0639 − 0.372i)21-s + 0.834·23-s + 0.931·25-s + (0.872 + 0.487i)27-s + 0.153i·29-s − 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.556870607\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.556870607\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 - 0.292i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 0.585iT - 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 - 2.24iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 0.828iT - 29T^{2} \) |
| 31 | \( 1 + 6.82iT - 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 6.48iT - 43T^{2} \) |
| 47 | \( 1 - 9.65T + 47T^{2} \) |
| 53 | \( 1 - 9.31iT - 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 + 5.75T + 61T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 8.58T + 83T^{2} \) |
| 89 | \( 1 + 3.65iT - 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−9.238918933191393368884044222865, −9.051594087955044777663994040781, −7.965686413598748645127029478433, −7.32596818286730315528240634969, −6.47274190361606185161668938533, −5.22098639693271642780626291486, −4.29799595698970107358947150230, −3.52217112842852372767720916966, −2.43736086931292882411768347360, −1.17377222708132401030632508982,
1.31887958695837183125099705606, 2.56235887463147169955024995061, 3.33698125326580778180454763063, 4.38620411280687931217697973946, 5.39352198867615987436534072099, 6.75970067428874770537691831875, 7.00932354781171283490275068874, 8.214184326852828690772416588645, 8.802662347771955045949803773413, 9.434745565021708967371343304200