L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (1 − i)7-s + (−0.707 + 0.707i)8-s − 1.41i·11-s + (1 + i)13-s + 1.41·14-s − 1.00·16-s + (1.00 − 1.00i)22-s + 1.41i·26-s + (1.00 + 1.00i)28-s + (−0.707 − 0.707i)32-s + (−1 + i)37-s − 1.41i·41-s + 1.41·44-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (1 − i)7-s + (−0.707 + 0.707i)8-s − 1.41i·11-s + (1 + i)13-s + 1.41·14-s − 1.00·16-s + (1.00 − 1.00i)22-s + 1.41i·26-s + (1.00 + 1.00i)28-s + (−0.707 − 0.707i)32-s + (−1 + i)37-s − 1.41i·41-s + 1.41·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.823883313\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.823883313\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1 + i)T - iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 53 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.208116113098935324590202048879, −8.576300286080600876179035433467, −7.918280237630170996361649557032, −7.15402655892983036841783193052, −6.31269381469828824786763980612, −5.62105468480300655623066643085, −4.56295965746085436971958449694, −3.97170366370086445782045915606, −3.05662780229486034017187702195, −1.44817319801647164540447848030,
1.52902223660804451088843312210, 2.32188695783414856389982357589, 3.41868319869982043926084036739, 4.47218056618065224555619094941, 5.20080844306601801970805807940, 5.81681196100221932068276425534, 6.82163482035063539648652382103, 7.88109279715263019649285873486, 8.682436847002515711980900592469, 9.475538939175700840627812522786