| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + 2.56i·7-s − i·8-s − 9-s + 2.56·11-s − i·12-s − 2i·13-s − 2.56·14-s + 16-s − 3.12i·17-s − i·18-s + 7.68·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.968i·7-s − 0.353i·8-s − 0.333·9-s + 0.772·11-s − 0.288i·12-s − 0.554i·13-s − 0.684·14-s + 0.250·16-s − 0.757i·17-s − 0.235i·18-s + 1.76·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911338867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911338867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2.56iT - 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 3.12iT - 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 + 1.43iT - 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 37 | \( 1 + 3.12iT - 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 + 12.8iT - 43T^{2} \) |
| 47 | \( 1 - 5.12iT - 47T^{2} \) |
| 53 | \( 1 + 7.43iT - 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 15.3iT - 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 4.31T + 79T^{2} \) |
| 83 | \( 1 + 14.2iT - 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.539874926205176287046220884749, −7.61850490335299997950121477703, −7.09685032518647667588844324737, −6.07693143808130644139031888613, −5.46939516065669946944520553818, −5.04121553231473481601930658221, −3.93805618749383463815397868393, −3.26092895629926949272166384959, −2.22282443393755612369413455493, −0.69339981519109724514374122871,
0.959054634563096058020569026454, 1.51204612730515605206824358824, 2.67415892373530616426787958422, 3.68828068135981655495204589273, 4.12487037964941697116975643156, 5.19643992305739386259010314904, 6.00799539325854842257781910292, 6.86435719330280045372485413156, 7.48924980060939509274298621947, 8.091775877390436927965937711106
