L(s) = 1 | + (0.674 − 2.51i)3-s + (−0.647 − 2.14i)5-s + (−0.839 + 1.45i)7-s + (−3.29 − 1.90i)9-s + (0.758 − 2.83i)11-s + (3.53 + 0.727i)13-s + (−5.82 + 0.186i)15-s + (−7.90 + 2.11i)17-s + (2.32 − 0.621i)19-s + (3.09 + 3.09i)21-s + (1.57 + 0.421i)23-s + (−4.16 + 2.77i)25-s + (−1.47 + 1.47i)27-s + (2.41 − 1.39i)29-s + (7.32 − 7.32i)31-s + ⋯ |
L(s) = 1 | + (0.389 − 1.45i)3-s + (−0.289 − 0.957i)5-s + (−0.317 + 0.549i)7-s + (−1.09 − 0.633i)9-s + (0.228 − 0.853i)11-s + (0.979 + 0.201i)13-s + (−1.50 + 0.0480i)15-s + (−1.91 + 0.513i)17-s + (0.532 − 0.142i)19-s + (0.675 + 0.675i)21-s + (0.328 + 0.0879i)23-s + (−0.832 + 0.554i)25-s + (−0.284 + 0.284i)27-s + (0.448 − 0.259i)29-s + (1.31 − 1.31i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.657443 - 1.09441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.657443 - 1.09441i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.647 + 2.14i)T \) |
| 13 | \( 1 + (-3.53 - 0.727i)T \) |
good | 3 | \( 1 + (-0.674 + 2.51i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (0.839 - 1.45i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.758 + 2.83i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (7.90 - 2.11i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.32 + 0.621i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.57 - 0.421i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.41 + 1.39i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.32 + 7.32i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.08 - 3.60i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.77 - 1.28i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.786 - 2.93i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + (6.66 + 6.66i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.16 + 4.35i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (2.16 - 3.75i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.28 - 2.47i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.63 - 6.10i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 - 7.13iT - 73T^{2} \) |
| 79 | \( 1 - 9.73iT - 79T^{2} \) |
| 83 | \( 1 - 6.67T + 83T^{2} \) |
| 89 | \( 1 + (-0.578 - 0.154i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (10.4 + 6.03i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.81729473162779023745371388284, −11.15611952542192364334435379633, −9.301356251646542060853805400732, −8.584587749249546127962967039865, −7.975822664793409104886996829129, −6.60783742917373654299248085626, −5.92432463165020916307226317510, −4.23099466216416176279850750215, −2.56403918919994340483857261456, −1.04711940603510689611019257416,
2.80689072892184153007011759440, 3.89720440611654403554415648343, 4.68763441396959567537809453029, 6.38815381369955657294775542703, 7.33073359778827816611782045952, 8.743917735363422959658341072308, 9.507947209286489352173350737557, 10.65530780119964096765345164977, 10.76709991438589976438113495591, 12.08450402192835693218764355473