L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.807 − 1.53i)3-s − 1.00i·4-s + (−2.02 − 0.947i)5-s + (1.65 + 0.512i)6-s + (−1.59 − 1.59i)7-s + (0.707 + 0.707i)8-s + (−1.69 + 2.47i)9-s + (2.10 − 0.762i)10-s + 2.58i·11-s + (−1.53 + 0.807i)12-s + (0.707 − 0.707i)13-s + 2.25·14-s + (0.183 + 3.86i)15-s − 1.00·16-s + (−0.155 + 0.155i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.465 − 0.884i)3-s − 0.500i·4-s + (−0.905 − 0.423i)5-s + (0.675 + 0.209i)6-s + (−0.603 − 0.603i)7-s + (0.250 + 0.250i)8-s + (−0.565 + 0.824i)9-s + (0.664 − 0.241i)10-s + 0.780i·11-s + (−0.442 + 0.232i)12-s + (0.196 − 0.196i)13-s + 0.603·14-s + (0.0472 + 0.998i)15-s − 0.250·16-s + (−0.0376 + 0.0376i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0804226 + 0.152621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0804226 + 0.152621i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.807 + 1.53i)T \) |
| 5 | \( 1 + (2.02 + 0.947i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (1.59 + 1.59i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.58iT - 11T^{2} \) |
| 17 | \( 1 + (0.155 - 0.155i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.77iT - 19T^{2} \) |
| 23 | \( 1 + (-1.81 - 1.81i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.890T + 29T^{2} \) |
| 31 | \( 1 + 9.39T + 31T^{2} \) |
| 37 | \( 1 + (2.88 + 2.88i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.11iT - 41T^{2} \) |
| 43 | \( 1 + (4.38 - 4.38i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.71 - 3.71i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.24 + 2.24i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.75T + 59T^{2} \) |
| 61 | \( 1 + 2.28T + 61T^{2} \) |
| 67 | \( 1 + (10.9 + 10.9i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.60iT - 71T^{2} \) |
| 73 | \( 1 + (1.65 - 1.65i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.1iT - 79T^{2} \) |
| 83 | \( 1 + (5.93 + 5.93i)T + 83iT^{2} \) |
| 89 | \( 1 - 6.43T + 89T^{2} \) |
| 97 | \( 1 + (-8.84 - 8.84i)T + 97iT^{2} \) |
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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
−11.62697100050132940690560892578, −10.74803502391919228086021390542, −9.779327539752921379141792995392, −8.609773389368522906352473456816, −7.65991757227827502053604181831, −7.20224568931207940999884027899, −6.13516583148325446550920863279, −5.02840466200727436257955196044, −3.63735793942549410888986211273, −1.50899237071781205411836436699,
0.14580006290439120433250589057, 2.89181918755306125328242085941, 3.68818095388681332962841681236, 4.94397686305750377789556549793, 6.25523265763313153540849497054, 7.25384266899798295426802536601, 8.738394541153570946311557395873, 9.051318932764702860741872396519, 10.27880629227957280937143731346, 11.04578298390662721771613128051