L(s) = 1 | + (−0.222 + 0.974i)2-s + (0.222 − 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.900 + 0.433i)6-s + (0.623 − 0.781i)8-s + (−0.900 − 0.433i)9-s + (−0.846 − 1.75i)11-s + (−0.623 + 0.781i)12-s + (0.623 + 0.781i)16-s + (−0.678 + 0.541i)17-s + (0.623 − 0.781i)18-s + (0.376 − 0.781i)19-s + (1.90 − 0.433i)22-s + (−0.623 − 0.781i)24-s + (−0.222 − 0.974i)25-s + ⋯ |
L(s) = 1 | + (−0.222 + 0.974i)2-s + (0.222 − 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.900 + 0.433i)6-s + (0.623 − 0.781i)8-s + (−0.900 − 0.433i)9-s + (−0.846 − 1.75i)11-s + (−0.623 + 0.781i)12-s + (0.623 + 0.781i)16-s + (−0.678 + 0.541i)17-s + (0.623 − 0.781i)18-s + (0.376 − 0.781i)19-s + (1.90 − 0.433i)22-s + (−0.623 − 0.781i)24-s + (−0.222 − 0.974i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7236049804\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7236049804\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.222 - 0.974i)T \) |
| 3 | \( 1 + (-0.222 + 0.974i)T \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
good | 5 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.846 + 1.75i)T + (-0.623 + 0.781i)T^{2} \) |
| 13 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (0.678 - 0.541i)T + (0.222 - 0.974i)T^{2} \) |
| 19 | \( 1 + (-0.376 + 0.781i)T + (-0.623 - 0.781i)T^{2} \) |
| 23 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 53 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 + (-1.52 + 1.21i)T + (0.222 - 0.974i)T^{2} \) |
| 61 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 + (-1.80 - 0.867i)T + (0.623 + 0.781i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1.90 - 0.433i)T + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 97 | \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \) |
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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
−9.733204993555613134951521996641, −8.645548679554119015281520359451, −8.375246232920054217936101233283, −7.53865475268469011225255330976, −6.61014690145896450058453425822, −5.96868260799295179570119997608, −5.17327599026101054382925913458, −3.75815495996373369189975912752, −2.50963545228650022668822852925, −0.71027303754854927997856788619,
2.00552000476695780850184317303, 2.93014581774759176988059150641, 4.08199971345956873925503613199, 4.76112673816464866143424879186, 5.56872506800682313843033253716, 7.27787983207685457910554441688, 7.980916137590000807677473383014, 8.990145152821753564642273040182, 9.739195432769842658308827047880, 10.06255839627028207195231002271