| L(s) = 1 | + (−1 − i)2-s + 2i·4-s + (−3 − 3i)7-s + (2 − 2i)8-s + 6i·14-s − 4·16-s + (−1 + i)23-s + (6 − 6i)28-s + 6i·29-s + (4 + 4i)32-s − 12·41-s + (−9 + 9i)43-s + 2·46-s + (−7 − 7i)47-s + 11i·49-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)2-s + i·4-s + (−1.13 − 1.13i)7-s + (0.707 − 0.707i)8-s + 1.60i·14-s − 16-s + (−0.208 + 0.208i)23-s + (1.13 − 1.13i)28-s + 1.11i·29-s + (0.707 + 0.707i)32-s − 1.87·41-s + (−1.37 + 1.37i)43-s + 0.294·46-s + (−1.02 − 1.02i)47-s + 1.57i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (3 + 3i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (1 - i)T - 23iT^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + (9 - 9i)T - 43iT^{2} \) |
| 47 | \( 1 + (7 + 7i)T + 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + (3 + 3i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (11 - 11i)T - 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−9.840646564602905882016952216415, −8.874163464935605254975256118737, −7.980327520811191375411657917353, −7.04592510983331967017197185623, −6.48625854283405323859461544741, −4.86774410197011933552504603467, −3.69955452081662181540677535696, −3.09261678250413111831336110729, −1.49522296593912016389884812796, 0,
1.97857098273931622214722382350, 3.22214226622701215711523678670, 4.76605875059416824221828872885, 5.79875495962519145827089414206, 6.36111142548848000335399366700, 7.22506277653226747992125334886, 8.339160821893038661546252725528, 8.882969361526073725695830923503, 9.783177142889602176615982615267
