L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)8-s + (−0.923 − 0.382i)9-s + (1.08 − 0.216i)11-s + (−0.707 − 0.707i)14-s − i·16-s + i·18-s + (−0.617 − 0.923i)22-s + (0.707 − 1.70i)23-s + (0.382 + 0.923i)25-s + (−0.382 + 0.923i)28-s + (−1.63 − 0.324i)29-s + (−0.923 + 0.382i)32-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)8-s + (−0.923 − 0.382i)9-s + (1.08 − 0.216i)11-s + (−0.707 − 0.707i)14-s − i·16-s + i·18-s + (−0.617 − 0.923i)22-s + (0.707 − 1.70i)23-s + (0.382 + 0.923i)25-s + (−0.382 + 0.923i)28-s + (−1.63 − 0.324i)29-s + (−0.923 + 0.382i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7217072909\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7217072909\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 + 0.923i)T \) |
| 7 | \( 1 + (-0.923 + 0.382i)T \) |
good | 3 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 5 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 11 | \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.382 - 0.0761i)T + (0.923 - 0.382i)T^{2} \) |
| 59 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 61 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \) |
| 71 | \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 83 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 89 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.30535608220783449927175487973, −10.42393094501860562727785627738, −9.227819817014893372036621675043, −8.695010867257894089787807378573, −7.77083985009566780699685828694, −6.56365523694485657660693774383, −5.12654783881691722145337481504, −4.07931343259714838553717245036, −2.94726948009999986478979072023, −1.37792746542501400546364602219,
1.80633320806324638069443438443, 3.85016667805516508285764558040, 5.17860359059096523011820989178, 5.75561863199073584781311589497, 7.04119703829106614924514546309, 7.80566665899997133383930424655, 8.892752301194280483722105221444, 9.199355743121946164231162082666, 10.62154583506694796293791672668, 11.33776025087097722817306897302