L(s) = 1 | + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + 11-s + (0.707 + 0.707i)13-s + 1.00i·15-s + (0.707 − 0.707i)17-s − 1.41i·19-s + 1.00i·25-s + (−0.707 + 0.707i)27-s − i·29-s − 1.41·31-s + (−0.707 − 0.707i)33-s − 1.00i·39-s + 1.41·41-s + (−1 − i)43-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + 11-s + (0.707 + 0.707i)13-s + 1.00i·15-s + (0.707 − 0.707i)17-s − 1.41i·19-s + 1.00i·25-s + (−0.707 + 0.707i)27-s − i·29-s − 1.41·31-s + (−0.707 − 0.707i)33-s − 1.00i·39-s + 1.41·41-s + (−1 − i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7658233254\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7658233254\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
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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
−9.198561741960579137813950150182, −8.310026909099812598915847133789, −7.34761003448283470360950283803, −6.81787762164204253608288583019, −6.01391923961920037128300571150, −5.11706896288769908397277154076, −4.20756965434041532849194256647, −3.37410373130829752072206870480, −1.71049858305065369401342424399, −0.67000012351427312704590870816,
1.55415969559917892806676621842, 3.30123222172558483690853473092, 3.78288295370457570616589227006, 4.69109466994569373153976826800, 5.84436109040130761113841306201, 6.17295131574672602007520387326, 7.38980825143633712539839792610, 7.975098927121648933740028735070, 8.862352547186727220491257408482, 9.871897364729658763499641042711