L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + 1.00i·6-s + (−2.63 + 0.189i)7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + 1.46·11-s + (−0.707 − 0.707i)12-s + (0.189 − 0.189i)13-s + (1.73 − 2i)14-s − 1.00·16-s + (−3.53 − 3.53i)17-s + (0.707 + 0.707i)18-s − 0.535·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + 0.408i·6-s + (−0.997 + 0.0716i)7-s + (0.250 + 0.250i)8-s − 0.333i·9-s + 0.441·11-s + (−0.204 − 0.204i)12-s + (0.0525 − 0.0525i)13-s + (0.462 − 0.534i)14-s − 0.250·16-s + (−0.857 − 0.857i)17-s + (0.166 + 0.166i)18-s − 0.122·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6732315700\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6732315700\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.189i)T \) |
good | 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + (-0.189 + 0.189i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.53 + 3.53i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.26iT - 29T^{2} \) |
| 31 | \( 1 + 5.92iT - 31T^{2} \) |
| 37 | \( 1 + (4.89 - 4.89i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.26iT - 41T^{2} \) |
| 43 | \( 1 + (0.189 + 0.189i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.76 + 8.76i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.39 + 7.39i)T + 53iT^{2} \) |
| 59 | \( 1 - 3.19T + 59T^{2} \) |
| 61 | \( 1 + 4.46iT - 61T^{2} \) |
| 67 | \( 1 + (10.1 - 10.1i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.53T + 71T^{2} \) |
| 73 | \( 1 + (-5.93 + 5.93i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.39iT - 79T^{2} \) |
| 83 | \( 1 + (4.57 - 4.57i)T - 83iT^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + (-11.9 - 11.9i)T + 97iT^{2} \) |
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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
−9.569239352162119616663383137162, −8.794520426249600782979200712713, −8.072005353133617370839819848716, −6.98860228431715994847760210739, −6.59759048945552298069561997534, −5.64111622739156794672766038045, −4.36721071814325537192498019149, −3.18440704779564779444308599372, −2.02143903838823030023239542793, −0.32775217798076536249618302901,
1.64174781137156115278535678415, 2.95796146022287404728338367227, 3.71511074113471462979507680920, 4.68288591482310508917966993982, 6.09782966285837653151526583837, 6.87309997382606613579090816718, 7.87131780835003688709483626590, 8.949289181977190913279793130026, 9.155318012082608018497786623372, 10.24002919525725996450334018922