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 − 0.585i·11-s + (−4 + 4i)13-s + 1.00·14-s − 1.00·16-s + (−0.585 + 0.585i)17-s + 2.82i·19-s + (−0.414 − 0.414i)22-s + (−4.82 − 4.82i)23-s + 5.65i·26-s + (0.707 − 0.707i)28-s + 0.828·29-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 − 0.176i·11-s + (−1.10 + 1.10i)13-s + 0.267·14-s − 0.250·16-s + (−0.142 + 0.142i)17-s + 0.648i·19-s + (−0.0883 − 0.0883i)22-s + (−1.00 − 1.00i)23-s + 1.10i·26-s + (0.133 − 0.133i)28-s + 0.153·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6123322223\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6123322223\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.585iT - 11T^{2} \) |
| 13 | \( 1 + (4 - 4i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.585 - 0.585i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + (4.82 + 4.82i)T + 23iT^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + (6.24 + 6.24i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.17iT - 41T^{2} \) |
| 43 | \( 1 + (6.07 - 6.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (9.24 - 9.24i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.58 - 2.58i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 + (8.41 + 8.41i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.17iT - 71T^{2} \) |
| 73 | \( 1 + (7.07 - 7.07i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.65iT - 79T^{2} \) |
| 83 | \( 1 + (-0.828 - 0.828i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.17T + 89T^{2} \) |
| 97 | \( 1 + (-4.58 - 4.58i)T + 97iT^{2} \) |
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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
−8.999669119563229774586159354840, −8.225398039033881296857734072802, −7.39278875850083342181389429511, −6.46979991395662020674358669529, −5.89759883913406181838949175106, −4.78112257973831550785790208981, −4.41852509076874349325921878710, −3.33197555375748029013213590785, −2.34874342955912613920407212635, −1.58988401323574838412331701364,
0.14407430777976863492772997338, 1.84912247292260337394285096207, 2.94703555562049082065346653831, 3.74372292793967583728040003755, 4.85506224957871488336247804520, 5.19420848483835656906849703847, 6.15526081538407633142898494522, 7.05316505003664333030627414489, 7.54876490870773672807114860941, 8.275897141276937566688650806880