L(s) = 1 | + 0.790·3-s + (0.139 + 0.139i)7-s − 2.37·9-s + (−2.94 + 2.94i)11-s − 0.235i·13-s + (−2.06 − 2.06i)17-s + (2.55 − 2.55i)19-s + (0.110 + 0.110i)21-s + (−4.62 + 4.62i)23-s − 4.24·27-s + (−6.66 − 6.66i)29-s + 3.43i·31-s + (−2.32 + 2.32i)33-s − 1.38i·37-s − 0.186i·39-s + ⋯ |
L(s) = 1 | + 0.456·3-s + (0.0528 + 0.0528i)7-s − 0.791·9-s + (−0.888 + 0.888i)11-s − 0.0653i·13-s + (−0.499 − 0.499i)17-s + (0.585 − 0.585i)19-s + (0.0241 + 0.0241i)21-s + (−0.964 + 0.964i)23-s − 0.817·27-s + (−1.23 − 1.23i)29-s + 0.616i·31-s + (−0.405 + 0.405i)33-s − 0.227i·37-s − 0.0298i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1167433399\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1167433399\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.790T + 3T^{2} \) |
| 7 | \( 1 + (-0.139 - 0.139i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.94 - 2.94i)T - 11iT^{2} \) |
| 13 | \( 1 + 0.235iT - 13T^{2} \) |
| 17 | \( 1 + (2.06 + 2.06i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.55 + 2.55i)T - 19iT^{2} \) |
| 23 | \( 1 + (4.62 - 4.62i)T - 23iT^{2} \) |
| 29 | \( 1 + (6.66 + 6.66i)T + 29iT^{2} \) |
| 31 | \( 1 - 3.43iT - 31T^{2} \) |
| 37 | \( 1 + 1.38iT - 37T^{2} \) |
| 41 | \( 1 + 8.26iT - 41T^{2} \) |
| 43 | \( 1 - 5.40iT - 43T^{2} \) |
| 47 | \( 1 + (6.84 - 6.84i)T - 47iT^{2} \) |
| 53 | \( 1 + 8.19T + 53T^{2} \) |
| 59 | \( 1 + (4.32 + 4.32i)T + 59iT^{2} \) |
| 61 | \( 1 + (9.15 - 9.15i)T - 61iT^{2} \) |
| 67 | \( 1 + 5.00iT - 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.44T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 + (-0.515 - 0.515i)T + 97iT^{2} \) |
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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
−9.484068310164045759380821331005, −9.258189643015077857623502909098, −7.936277046877060798125782886154, −7.74834994187408203363751920317, −6.64823453425323934750186438645, −5.60028628950691899072497379521, −4.93827979751318517509378679802, −3.80080431129391359678062405273, −2.77718948613791367739390903954, −1.95312023790929617870936795504,
0.03808557841365121469338334543, 1.86614033050442991502776741362, 2.96005075322870162813158160268, 3.69087695084349918983519135723, 4.92130025675730643339807124017, 5.78758283472707908984155704430, 6.46152972416606771720386378593, 7.78676291900079030707258992177, 8.122642137741297286118160835836, 8.931558738505564963965162923903