L(s) = 1 | + (0.913 − 0.406i)5-s − 0.209·7-s + (−0.809 − 0.587i)9-s + (0.169 − 0.122i)11-s + (−0.604 − 1.86i)17-s + (0.309 + 0.951i)19-s + (1.30 − 0.951i)23-s + (0.669 − 0.743i)25-s + (−0.190 + 0.0850i)35-s + 1.33·43-s + (−0.978 − 0.207i)45-s + (−0.309 + 0.951i)47-s − 0.956·49-s + (0.104 − 0.181i)55-s + (−1.08 + 0.786i)61-s + ⋯ |
L(s) = 1 | + (0.913 − 0.406i)5-s − 0.209·7-s + (−0.809 − 0.587i)9-s + (0.169 − 0.122i)11-s + (−0.604 − 1.86i)17-s + (0.309 + 0.951i)19-s + (1.30 − 0.951i)23-s + (0.669 − 0.743i)25-s + (−0.190 + 0.0850i)35-s + 1.33·43-s + (−0.978 − 0.207i)45-s + (−0.309 + 0.951i)47-s − 0.956·49-s + (0.104 − 0.181i)55-s + (−1.08 + 0.786i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.230628257\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230628257\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.913 + 0.406i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + 0.209T + T^{2} \) |
| 11 | \( 1 + (-0.169 + 0.122i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.604 + 1.86i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - 1.33T + T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−9.168620068137830440401217263893, −8.855853679953283466692252594252, −7.74321867160411728119097302640, −6.74251915974446348885405839760, −6.12986602211642625729575313661, −5.28840486892329587649307669664, −4.54510092321117087751966476915, −3.18233632467201119368798440317, −2.46892298113747544709107222301, −0.956568679633846329940462433716,
1.64496422860681170422904918114, 2.62120132765566967346987971891, 3.52322342757056530851500717599, 4.79848120215114808085245127554, 5.58929483002941659668875152144, 6.31126423671359152931352628834, 7.03416549840858707651559993398, 8.004182237893729903863062206207, 8.903480321813861435464527176956, 9.384942045391120625539208916536