L(s) = 1 | + (0.707 − 0.707i)3-s + (1 + i)5-s + 1.41i·7-s − 1.00i·9-s + 1.41·15-s + (1.00 + 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s + (1 − i)29-s − 1.41·31-s + (−1.41 + 1.41i)35-s + (1.00 − 1.00i)45-s − 1.00·49-s + (1 + i)53-s + (1.41 + 1.41i)59-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s + (1 + i)5-s + 1.41i·7-s − 1.00i·9-s + 1.41·15-s + (1.00 + 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s + (1 − i)29-s − 1.41·31-s + (−1.41 + 1.41i)35-s + (1.00 − 1.00i)45-s − 1.00·49-s + (1 + i)53-s + (1.41 + 1.41i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.887843188\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.887843188\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (-1 - i)T + iT^{2} \) |
| 7 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−8.882882662649836680002026042814, −8.317700262584930625241011855237, −7.32772956605410098486816404949, −6.70674809576638280568441831572, −5.91670545090749496973089760558, −5.53244551160912387720814522941, −4.05031999766707573593851654903, −2.81839659550972779658887060070, −2.54116998413610041546289081331, −1.63983652320964928946055910882,
1.18769800524870273713950342010, 2.17995600944609962171415975583, 3.42409462183955206607210737542, 4.13468988420988663562583351683, 4.96602860971618869624204441395, 5.48624913155813880209990266209, 6.69630270031417476892637764803, 7.41955003435143033852508055462, 8.320892085300359304247127877211, 8.878253786904448552981271102384