L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + (−0.253 − 2.22i)5-s + 1.00·6-s + (−2.47 − 2.47i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−1.75 − 1.39i)10-s + 2.74·11-s + (0.707 − 0.707i)12-s + (−1.20 − 1.20i)13-s − 3.50·14-s + (1.39 − 1.75i)15-s − 1.00·16-s + (−4.87 − 4.87i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (−0.113 − 0.993i)5-s + 0.408·6-s + (−0.935 − 0.935i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.553 − 0.440i)10-s + 0.827·11-s + (0.204 − 0.204i)12-s + (−0.333 − 0.333i)13-s − 0.935·14-s + (0.359 − 0.451i)15-s − 0.250·16-s + (−1.18 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.937540 - 1.48125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.937540 - 1.48125i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.253 + 2.22i)T \) |
| 19 | \( 1 + (-1.94 - 3.90i)T \) |
good | 7 | \( 1 + (2.47 + 2.47i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 + (1.20 + 1.20i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.87 + 4.87i)T + 17iT^{2} \) |
| 23 | \( 1 + (-0.0321 + 0.0321i)T - 23iT^{2} \) |
| 29 | \( 1 - 6.50T + 29T^{2} \) |
| 31 | \( 1 + 6.50iT - 31T^{2} \) |
| 37 | \( 1 + (-4.58 + 4.58i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.96iT - 41T^{2} \) |
| 43 | \( 1 + (-5.39 + 5.39i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.66 - 3.66i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.97 - 8.97i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.42T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 + (-7.00 + 7.00i)T - 67iT^{2} \) |
| 71 | \( 1 - 5.56iT - 71T^{2} \) |
| 73 | \( 1 + (2.19 - 2.19i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.225T + 79T^{2} \) |
| 83 | \( 1 + (3.87 - 3.87i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.13T + 89T^{2} \) |
| 97 | \( 1 + (8.76 - 8.76i)T - 97iT^{2} \) |
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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
−10.35583991413555462256292125812, −9.561531865545342634127698520275, −9.088323947260070409038194971800, −7.82497094471476111520329602991, −6.78628737978260857684018331667, −5.63269846947900325583454935186, −4.40616665660377039316255096222, −3.93146002372103382837105097397, −2.62132950640783382585414633104, −0.817992116117838225306405951640,
2.32909964388104243270299691597, 3.19673840315487413941278292501, 4.30388044753070360899147678224, 5.84144645147048142046789678325, 6.67953540565317928670373818288, 6.98593578799321053938051347850, 8.402619739674243467640297081315, 9.045924565386849100924009575507, 10.03607012131449784383612948851, 11.22606921005537573508675722696