L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + 3.41i·11-s + (1.43 + 1.43i)13-s + 1.00·14-s − 1.00·16-s + (1.54 + 1.54i)17-s − 2.04i·19-s + (−2.41 + 2.41i)22-s + (1.15 − 1.15i)23-s + 2.03i·26-s + (0.707 + 0.707i)28-s + 8.04·29-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + 1.02i·11-s + (0.399 + 0.399i)13-s + 0.267·14-s − 0.250·16-s + (0.374 + 0.374i)17-s − 0.470i·19-s + (−0.514 + 0.514i)22-s + (0.241 − 0.241i)23-s + 0.399i·26-s + (0.133 + 0.133i)28-s + 1.49·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.367930246\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.367930246\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 - 3.41iT - 11T^{2} \) |
| 13 | \( 1 + (-1.43 - 1.43i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.54 - 1.54i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.04iT - 19T^{2} \) |
| 23 | \( 1 + (-1.15 + 1.15i)T - 23iT^{2} \) |
| 29 | \( 1 - 8.04T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 + (0.0498 - 0.0498i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.94iT - 41T^{2} \) |
| 43 | \( 1 + (-4.65 - 4.65i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.39 - 1.39i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.97 - 2.97i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.25T + 59T^{2} \) |
| 61 | \( 1 + 6.88T + 61T^{2} \) |
| 67 | \( 1 + (5.84 - 5.84i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.92iT - 71T^{2} \) |
| 73 | \( 1 + (-5.97 - 5.97i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.6iT - 79T^{2} \) |
| 83 | \( 1 + (-1.52 + 1.52i)T - 83iT^{2} \) |
| 89 | \( 1 - 5.72T + 89T^{2} \) |
| 97 | \( 1 + (9.64 - 9.64i)T - 97iT^{2} \) |
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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
−8.843058318605898012027711949054, −7.937997371252991438517879282689, −7.39960954366621599159434687893, −6.59394387852837539444900971866, −5.99331825399816553681238279242, −4.85615262580955335260510234835, −4.49830013401196710822783603625, −3.50952805597037217430825853028, −2.48798244637996470348067801365, −1.30617622343175463711703285392,
0.65507076169128815026289976476, 1.80628484027739147500064729673, 2.95552284891903965096566697286, 3.55265849299716031694199798624, 4.53560309829002885037503376506, 5.49801464230711334551686652917, 5.87008464224038805233185788107, 6.86213890280736658613942733895, 7.78059928596646795093285837574, 8.562635830035805185500825576818