| L(s) = 1 | − 0.456i·2-s + i·3-s + 1.79·4-s + (2 + i)5-s + 0.456·6-s + 0.913i·7-s − 1.73i·8-s − 9-s + (0.456 − 0.913i)10-s + 1.79i·12-s + 0.913i·13-s + 0.417·14-s + (−1 + 2i)15-s + 2.79·16-s + 7.02i·17-s + 0.456i·18-s + ⋯ |
| L(s) = 1 | − 0.323i·2-s + 0.577i·3-s + 0.895·4-s + (0.894 + 0.447i)5-s + 0.186·6-s + 0.345i·7-s − 0.612i·8-s − 0.333·9-s + (0.144 − 0.288i)10-s + 0.517i·12-s + 0.253i·13-s + 0.111·14-s + (−0.258 + 0.516i)15-s + 0.697·16-s + 1.70i·17-s + 0.107i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.456526466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.456526466\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 - i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.456iT - 2T^{2} \) |
| 7 | \( 1 - 0.913iT - 7T^{2} \) |
| 13 | \( 1 - 0.913iT - 13T^{2} \) |
| 17 | \( 1 - 7.02iT - 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 - 0.582iT - 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 - 2.58T + 31T^{2} \) |
| 37 | \( 1 - 9.58iT - 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 4.37iT - 43T^{2} \) |
| 47 | \( 1 - 6.58iT - 47T^{2} \) |
| 53 | \( 1 + 5iT - 53T^{2} \) |
| 59 | \( 1 - 3.58T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 + 14.7iT - 67T^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 + 15.6iT - 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 - 12.2iT - 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 3.58iT - 97T^{2} \) |
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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
−9.613916301526506075783733368558, −8.744602707494548909963522821475, −7.912348039883231357643704726854, −6.75487103690445357925743615096, −6.20062183077121757660224346435, −5.58682326091720738960328554392, −4.30383098442226818716585037965, −3.37272544253470318171732258610, −2.38514807547878863481858209834, −1.64922955421527128658314865233,
0.872169443181384994440299486955, 2.14376968140260901998851661103, 2.73349674364345551979426968512, 4.25792188718505102566124520435, 5.44983194920763721162220063598, 5.88827769206638900969167950568, 6.91960571570504460687766867554, 7.26831728736222446694607900530, 8.269456073710762921090524588208, 9.012064706984668103239431095489
