| L(s) = 1 | + 0.840i·3-s + 4.04i·5-s + 3.59·7-s + 2.29·9-s − 4.29i·11-s + 1.77i·13-s − 3.40·15-s + 2.86·17-s + i·19-s + 3.01i·21-s − 2.74·23-s − 11.3·25-s + 4.44i·27-s + 2.29i·29-s − 2.33·31-s + ⋯ |
| L(s) = 1 | + 0.485i·3-s + 1.81i·5-s + 1.35·7-s + 0.764·9-s − 1.29i·11-s + 0.491i·13-s − 0.878·15-s + 0.695·17-s + 0.229i·19-s + 0.658i·21-s − 0.571·23-s − 2.27·25-s + 0.856i·27-s + 0.425i·29-s − 0.418·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33405 + 1.17136i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33405 + 1.17136i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 3 | \( 1 - 0.840iT - 3T^{2} \) |
| 5 | \( 1 - 4.04iT - 5T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 11 | \( 1 + 4.29iT - 11T^{2} \) |
| 13 | \( 1 - 1.77iT - 13T^{2} \) |
| 17 | \( 1 - 2.86T + 17T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 - 2.29iT - 29T^{2} \) |
| 31 | \( 1 + 2.33T + 31T^{2} \) |
| 37 | \( 1 + 8.06iT - 37T^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 + 0.241iT - 43T^{2} \) |
| 47 | \( 1 - 0.191T + 47T^{2} \) |
| 53 | \( 1 - 8.10iT - 53T^{2} \) |
| 59 | \( 1 - 4.36iT - 59T^{2} \) |
| 61 | \( 1 + 15.3iT - 61T^{2} \) |
| 67 | \( 1 + 13.2iT - 67T^{2} \) |
| 71 | \( 1 + 7.44T + 71T^{2} \) |
| 73 | \( 1 - 9.20T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 - 1.94iT - 83T^{2} \) |
| 89 | \( 1 + 3.24T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−10.87800034929086269275105162007, −10.26328687321368683247550140148, −9.224903156982549172667396217880, −8.014840774363933526002403763991, −7.38118896261539807749315023535, −6.37249759089813432884854477963, −5.38927422291231337617470827954, −4.07983359303476537914551628881, −3.24398596495062173932723258006, −1.84180597336912794802290229136,
1.16945431052930227571970127732, 1.92174850061769079945512990723, 4.22558577173644373860196504692, 4.80054779287385006873254849455, 5.60065942051576455821170339211, 7.13764875933003894698465580263, 7.957331294284799313829346444850, 8.445167857342402211959785247994, 9.591771012918944524164558402306, 10.23747021042337087730775560467
