L(s) = 1 | + i·3-s − 3.41i·5-s − 0.585·7-s − 9-s − 2i·11-s − 2.82i·13-s + 3.41·15-s − 7.65·17-s + 5.65i·19-s − 0.585i·21-s − 6.82·23-s − 6.65·25-s − i·27-s + 3.41i·29-s + 7.41·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.52i·5-s − 0.221·7-s − 0.333·9-s − 0.603i·11-s − 0.784i·13-s + 0.881·15-s − 1.85·17-s + 1.29i·19-s − 0.127i·21-s − 1.42·23-s − 1.33·25-s − 0.192i·27-s + 0.634i·29-s + 1.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2886256635\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2886256635\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 + 3.41iT - 5T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 - 3.41iT - 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + 1.65iT - 37T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 - 9.65iT - 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 7.89iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 1.65iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−8.970941737777286984655113769431, −8.404533476126845306789899189841, −7.81935153229061221671272701544, −6.28845110085632083970599126935, −5.75635496494063551230845869641, −4.69529917817382954550812740375, −4.20922803543672862546530609399, −3.02867836740251332675597581298, −1.57376168064719150215121700835, −0.10594265251544755500363837976,
2.09885868624070547911338808595, 2.59062824636754290046267531710, 3.85982861771932665568151632930, 4.78458463444902388236506512588, 6.29218109487093422444030427768, 6.62556673608918258678822105816, 7.17750601630547333007111173822, 8.145022381874293686001303433513, 9.073554141244365841250157506086, 9.935127265165520561833333282595