| L(s) = 1 | − 10i·3-s − 73·9-s − 18i·11-s + 90·17-s − 106i·19-s + 125·25-s + 460i·27-s − 180·33-s + 522·41-s − 290i·43-s − 343·49-s − 900i·51-s − 1.06e3·57-s + 846i·59-s + 70i·67-s + ⋯ |
| L(s) = 1 | − 1.92i·3-s − 2.70·9-s − 0.493i·11-s + 1.28·17-s − 1.27i·19-s + 25-s + 3.27i·27-s − 0.949·33-s + 1.98·41-s − 1.02i·43-s − 49-s − 2.47i·51-s − 2.46·57-s + 1.86i·59-s + 0.127i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.486717 - 1.17503i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.486717 - 1.17503i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 10iT - 27T^{2} \) |
| 5 | \( 1 - 125T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 + 18iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 90T + 4.91e3T^{2} \) |
| 19 | \( 1 + 106iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 - 522T + 6.89e4T^{2} \) |
| 43 | \( 1 + 290iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 - 846iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 70iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 430T + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.35e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.91e3T + 9.12e5T^{2} \) |
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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
−13.79389959274483022265465630356, −12.90132426249742083109406977213, −12.03337081396510708289117365251, −10.97771469654342395092919151371, −8.959211022354780171672446531596, −7.81880507298360077924760881446, −6.83429645786201923987309810735, −5.61031861032160692136242509158, −2.77936884203807191361072008305, −0.932458728169735447319813444621,
3.29405402020149563036686052772, 4.59799384083857088294902209489, 5.82122708255459669917213498215, 8.062515154678385557104580019096, 9.414049228270535961069706926038, 10.14580158709936650213664712947, 11.14588653674985863156697883957, 12.42256376614181800356557278129, 14.36572830236087905033820252125, 14.75283237125012775948700013759
