| L(s) = 1 | + i·2-s − 4-s + (−1.41 − 2.23i)7-s − i·8-s − 1.41i·11-s − 0.926i·13-s + (2.23 − 1.41i)14-s + 16-s + 2.23·17-s + 7.63i·19-s + 1.41·22-s − i·23-s + 0.926·26-s + (1.41 + 2.23i)28-s − 0.757i·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.534 − 0.845i)7-s − 0.353i·8-s − 0.426i·11-s − 0.256i·13-s + (0.597 − 0.377i)14-s + 0.250·16-s + 0.542·17-s + 1.75i·19-s + 0.301·22-s − 0.208i·23-s + 0.181·26-s + (0.267 + 0.422i)28-s − 0.140i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7288264012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7288264012\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
| good | 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 0.926iT - 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 - 7.63iT - 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 + 0.757iT - 29T^{2} \) |
| 31 | \( 1 + 4.08iT - 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 + 8.56T + 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 + 1.30T + 47T^{2} \) |
| 53 | \( 1 + 8.07iT - 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 + 0.926iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 + 13.9iT - 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 2.61T + 89T^{2} \) |
| 97 | \( 1 + 0.542iT - 97T^{2} \) |
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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
−8.188275114530723072196911275421, −7.81817141117772165519302004019, −7.00407000632604909127265485771, −6.17165523003886765104525325290, −5.69813076220175662960881392103, −4.63269640570609542457989081746, −3.78240000460980965478982102691, −3.13903070876658904205211244907, −1.53817887704575025300395803655, −0.23740602141061827367530712823,
1.29897959054539869270647516556, 2.50416316478804066920834252900, 3.04080502365365035679623546003, 4.13380815386094486888776045879, 4.99018836213390595152645383153, 5.67761078710600727272313730915, 6.67404776079536306849833836551, 7.31804616516561759538613391640, 8.440747025870879116810315397841, 8.963032869897052690446903871958
