L(s) = 1 | + 1.41i·5-s − 1.41i·17-s − 1.00·25-s − 1.41i·29-s − 2·37-s + 1.41i·41-s − 49-s − 1.41i·53-s + 2·61-s + 2.00·85-s + 1.41i·89-s + 1.41i·101-s + 1.41i·113-s + ⋯ |
L(s) = 1 | + 1.41i·5-s − 1.41i·17-s − 1.00·25-s − 1.41i·29-s − 2·37-s + 1.41i·41-s − 49-s − 1.41i·53-s + 2·61-s + 2.00·85-s + 1.41i·89-s + 1.41i·101-s + 1.41i·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7631466946\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7631466946\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 2T + T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.82364005276430261084917977675, −11.27272277549021617532348965545, −10.25421817631914371970215249127, −9.535470688748915098289904505943, −8.166265951663927993449620354249, −7.12730921247913974111547727651, −6.43717882936818947271319041241, −5.08820100691272879420543353273, −3.55301977274253424234910698938, −2.45752926428802070639318860267,
1.62477353788283076547995185246, 3.68653672110429161655706706768, 4.86655032832512391444028729382, 5.77387586277352677717732704340, 7.12516772822442984436426752071, 8.454377428287220388815227524639, 8.832238725349060277056030545343, 10.04676772816403246798420110897, 10.98983923879844934652412658938, 12.31149488754909459196760349043