L(s) = 1 | + (−1.75 − 0.950i)2-s + (2.19 + 3.34i)4-s + 3.98·7-s + (−0.677 − 7.97i)8-s + 8.39i·11-s + 9.77i·13-s + (−7.02 − 3.79i)14-s + (−6.38 + 14.6i)16-s − 12.1i·17-s + 17.3i·19-s + (7.97 − 14.7i)22-s − 2.97·23-s + (9.28 − 17.1i)26-s + (8.74 + 13.3i)28-s − 51.6·29-s + ⋯ |
L(s) = 1 | + (−0.879 − 0.475i)2-s + (0.548 + 0.836i)4-s + 0.569·7-s + (−0.0847 − 0.996i)8-s + 0.763i·11-s + 0.751i·13-s + (−0.501 − 0.270i)14-s + (−0.399 + 0.916i)16-s − 0.714i·17-s + 0.914i·19-s + (0.362 − 0.671i)22-s − 0.129·23-s + (0.357 − 0.661i)26-s + (0.312 + 0.476i)28-s − 1.78·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7750216555\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7750216555\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.75 + 0.950i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.98T + 49T^{2} \) |
| 11 | \( 1 - 8.39iT - 121T^{2} \) |
| 13 | \( 1 - 9.77iT - 169T^{2} \) |
| 17 | \( 1 + 12.1iT - 289T^{2} \) |
| 19 | \( 1 - 17.3iT - 361T^{2} \) |
| 23 | \( 1 + 2.97T + 529T^{2} \) |
| 29 | \( 1 + 51.6T + 841T^{2} \) |
| 31 | \( 1 - 30.7iT - 961T^{2} \) |
| 37 | \( 1 + 19.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 42.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 39.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 21.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 94.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 132. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 77.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 48.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 140.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 18.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 15iT - 9.40e3T^{2} \) |
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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
−9.986869929574981245272943885008, −9.394039588123328477680426355832, −8.605895295667281289236930782555, −7.59879767757194388026524387421, −7.14083366336773503197960507923, −5.93175053362285852910919979602, −4.63636757830927318428570978167, −3.67287673110957980631947561590, −2.31543428112716663459088635692, −1.41023067574342611203349226828,
0.34245671569983475365571538568, 1.67719729717906709807838723634, 2.98029219110878402041236064568, 4.47887967962574946259493717080, 5.65006008855438557407691828133, 6.17037558588293536382863554546, 7.48389462783718529030455136210, 7.902990913457468959193668995048, 8.865577845640099918484306958750, 9.462329962552713683508592297713