L(s) = 1 | − 2-s + (−0.707 + 1.58i)3-s + 4-s + (−1.41 − 1.73i)5-s + (0.707 − 1.58i)6-s − 8-s + (−2.00 − 2.23i)9-s + (1.41 + 1.73i)10-s − 4.68i·11-s + (−0.707 + 1.58i)12-s + 1.04·13-s + (3.73 − 1.01i)15-s + 16-s + 3.16i·17-s + (2.00 + 2.23i)18-s − 1.43i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.408 + 0.912i)3-s + 0.5·4-s + (−0.632 − 0.774i)5-s + (0.288 − 0.645i)6-s − 0.353·8-s + (−0.666 − 0.745i)9-s + (0.447 + 0.547i)10-s − 1.41i·11-s + (−0.204 + 0.456i)12-s + 0.289·13-s + (0.965 − 0.261i)15-s + 0.250·16-s + 0.766i·17-s + (0.471 + 0.527i)18-s − 0.328i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1793805644\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1793805644\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.707 - 1.58i)T \) |
| 5 | \( 1 + (1.41 + 1.73i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4.68iT - 11T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 17 | \( 1 - 3.16iT - 17T^{2} \) |
| 19 | \( 1 + 1.43iT - 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 6.62iT - 31T^{2} \) |
| 37 | \( 1 - 2.66iT - 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 - 11.2iT - 47T^{2} \) |
| 53 | \( 1 + 5T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 14.2iT - 67T^{2} \) |
| 71 | \( 1 - 6.92iT - 71T^{2} \) |
| 73 | \( 1 + 3.50T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.06iT - 83T^{2} \) |
| 89 | \( 1 + 4.91T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−9.697409835186209025058145920919, −9.083291651286309346268464307028, −8.306408214684404565508277566004, −7.900010737852122011202314013528, −6.36994404504282803190808941410, −5.87494491046301245595924390260, −4.78852060450837740037014784665, −3.89764317695389921583438192470, −3.03251288289690600165516106845, −1.15419089672038930816474269078,
0.10704242790980751892613349815, 1.74120294002365501975768635007, 2.62289294747781303989990975900, 3.90539796840600684467107814523, 5.15766458728008383897136352120, 6.23339235810092240941623795595, 6.98770345403079331936077834075, 7.47563619357259825693865521762, 8.100071967681729272162317614651, 9.113173014142529157816015059592