L(s) = 1 | + (−0.923 − 0.382i)3-s + (−0.382 − 0.923i)5-s + (0.707 + 0.707i)9-s + i·15-s + 0.765i·17-s + (−0.707 + 1.70i)19-s + (−0.541 − 0.541i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s − 1.41·31-s + (0.382 − 0.923i)45-s + 1.84i·47-s − i·49-s + (0.292 − 0.707i)51-s + (−1.30 + 0.541i)53-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)3-s + (−0.382 − 0.923i)5-s + (0.707 + 0.707i)9-s + i·15-s + 0.765i·17-s + (−0.707 + 1.70i)19-s + (−0.541 − 0.541i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s − 1.41·31-s + (0.382 − 0.923i)45-s + 1.84i·47-s − i·49-s + (0.292 − 0.707i)51-s + (−1.30 + 0.541i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4290772590\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4290772590\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.923 + 0.382i)T \) |
| 5 | \( 1 + (0.382 + 0.923i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 - 0.765iT - T^{2} \) |
| 19 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 - 1.84iT - T^{2} \) |
| 53 | \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 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
−8.630237730020813806907480962631, −8.037204238832448877373837570036, −7.47554319625722242896424193145, −6.43009989875747999478729950173, −5.86744281125613321433721395568, −5.19294093708621070422109194480, −4.27481345635356732150143947334, −3.74179661941220416939354769724, −2.06797710748769756728092611893, −1.27083398269622641323926772097,
0.28780490475417867853645958928, 2.03324661328565186240204089561, 3.13618023440832369682196949354, 3.93624417229941060086304764180, 4.76398116278549248463651634622, 5.50030035548003184541349972186, 6.37735651383471689990163521201, 6.99043545873165237413718947610, 7.46446238096068360728587044963, 8.565281141776478860160783225745