L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)7-s − 1.73i·11-s + (1.22 − 1.22i)13-s + (1.22 + 1.22i)17-s − 1.00·21-s + (0.707 − 0.707i)27-s + i·29-s + (1.22 − 1.22i)33-s + 1.73·39-s + (−0.707 + 0.707i)47-s − 1.00i·49-s + 1.73i·51-s + (1.22 + 1.22i)77-s − 1.73·79-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)7-s − 1.73i·11-s + (1.22 − 1.22i)13-s + (1.22 + 1.22i)17-s − 1.00·21-s + (0.707 − 0.707i)27-s + i·29-s + (1.22 − 1.22i)33-s + 1.73·39-s + (−0.707 + 0.707i)47-s − 1.00i·49-s + 1.73i·51-s + (1.22 + 1.22i)77-s − 1.73·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.573288333\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.573288333\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 17 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
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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
−8.846219724384135740165775571266, −8.452355178425761480007928095042, −7.926124554934643042621153434215, −6.45736934246057966214138358867, −5.91618511466601041479867570728, −5.36331141832873474550629075640, −3.84841481360405129705080058685, −3.34176718332348448457018925677, −2.92361612840620591133113837653, −1.14110438759982320806246556017,
1.31861371445721950254959449316, 2.20459059508061100604739136351, 3.22708116905437333423136752491, 4.12809404009291944711259580417, 4.92042386509984541600983611206, 6.14179227314999565183151230741, 6.99405597197754309683610617723, 7.30244347862870413999360369309, 8.045443063098225976184112796710, 8.966882516786987427090863452908