L(s) = 1 | + (−0.382 + 0.923i)3-s + (−0.382 + 0.923i)5-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (1.30 + 1.30i)13-s + (−0.707 − 0.707i)15-s + 0.765i·19-s + (−0.382 − 0.923i)21-s + (−1 + i)23-s + (−0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s + (−0.382 − 0.923i)35-s + (−1.70 + 0.707i)39-s + (0.923 − 0.382i)45-s − 1.00i·49-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)3-s + (−0.382 + 0.923i)5-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (1.30 + 1.30i)13-s + (−0.707 − 0.707i)15-s + 0.765i·19-s + (−0.382 − 0.923i)21-s + (−1 + i)23-s + (−0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s + (−0.382 − 0.923i)35-s + (−1.70 + 0.707i)39-s + (0.923 − 0.382i)45-s − 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6776676954\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6776676954\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - 0.765iT - T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 + 1.84T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.291606753473069117061142257734, −8.628466760994132147420550401065, −7.75676114419796068298993820708, −6.67848441892336359934626008749, −6.16131955868708788937691875931, −5.64395187953557584981783520288, −4.38218632501636976960235559426, −3.69973904004294307132832688393, −3.15820378983258923510131805670, −1.89598614928410381854738811445,
0.44966988908178089992122292879, 1.32075008455084012930693112558, 2.72254477392863742905442347963, 3.69627059804203428995252987243, 4.56529343872111335296497579754, 5.56123034048234244032175717241, 6.13306125566456597371095945117, 6.91441653393674881836856050193, 7.72104200396093768400290997969, 8.304788075282511357065140972340