L(s) = 1 | + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s − 0.618i·7-s + (−0.309 − 0.951i)9-s + (0.951 − 0.309i)12-s + (1.53 − 0.5i)13-s + (0.309 + 0.951i)16-s + (−1.30 + 0.951i)19-s + (−0.500 − 0.363i)21-s + (−0.951 − 0.309i)27-s + (0.363 − 0.500i)28-s + (−0.5 + 0.363i)31-s + (0.309 − 0.951i)36-s + (−1.53 + 0.5i)37-s + (0.5 − 1.53i)39-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s − 0.618i·7-s + (−0.309 − 0.951i)9-s + (0.951 − 0.309i)12-s + (1.53 − 0.5i)13-s + (0.309 + 0.951i)16-s + (−1.30 + 0.951i)19-s + (−0.500 − 0.363i)21-s + (−0.951 − 0.309i)27-s + (0.363 − 0.500i)28-s + (−0.5 + 0.363i)31-s + (0.309 − 0.951i)36-s + (−1.53 + 0.5i)37-s + (0.5 − 1.53i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.695464957\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.695464957\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.587 + 0.809i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + 0.618iT - T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61iT - T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
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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
−8.952664253373064024502681254337, −8.377433676952144338937357727943, −7.85222485413483423116955888184, −6.93131430814931482509878539484, −6.47076885798098651715884181279, −5.59656060646979222424092467627, −3.80988417751737619499013416945, −3.57581295295140894388646683057, −2.29833845798923823362260250907, −1.38014072890866274486869579556,
1.73872841089749654340073733757, 2.59395716920068404103478552417, 3.60576401646659514413972763585, 4.56830205196087352184927622167, 5.54352531577112260075978216585, 6.24603497359428433789404141246, 7.05382115478427860382521822736, 8.179593299426876601082867547904, 8.837566921150861938477894180746, 9.390384338638181464168014582578