| L(s) = 1 | − 8.48i·5-s − 7i·7-s − 8.48·11-s + 19.0i·13-s − 14.6·17-s + 8.66·19-s + 14.6i·23-s − 46.9·25-s − 50.9i·29-s + 10i·31-s − 59.3·35-s − 19.0i·37-s − 58.7·41-s − 41.5·43-s + 73.4i·47-s + ⋯ |
| L(s) = 1 | − 1.69i·5-s − i·7-s − 0.771·11-s + 1.46i·13-s − 0.864·17-s + 0.455·19-s + 0.638i·23-s − 1.87·25-s − 1.75i·29-s + 0.322i·31-s − 1.69·35-s − 0.514i·37-s − 1.43·41-s − 0.966·43-s + 1.56i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03926360956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03926360956\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.48iT - 25T^{2} \) |
| 7 | \( 1 + 7iT - 49T^{2} \) |
| 11 | \( 1 + 8.48T + 121T^{2} \) |
| 13 | \( 1 - 19.0iT - 169T^{2} \) |
| 17 | \( 1 + 14.6T + 289T^{2} \) |
| 19 | \( 1 - 8.66T + 361T^{2} \) |
| 23 | \( 1 - 14.6iT - 529T^{2} \) |
| 29 | \( 1 + 50.9iT - 841T^{2} \) |
| 31 | \( 1 - 10iT - 961T^{2} \) |
| 37 | \( 1 + 19.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 58.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 41.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 73.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 25.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 19.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 116.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 71T + 5.32e3T^{2} \) |
| 79 | \( 1 - 151iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 135.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 161.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 25T + 9.40e3T^{2} \) |
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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
−9.394271381725794280264221024423, −8.580144915527969230418619675166, −7.893485551729538963474454363548, −7.08846826126842525542159405545, −6.10600492460209463432530343746, −5.03662355382126963103080774592, −4.50085818396554233168193427347, −3.76660575400644437269900305099, −2.12010177178148880065823325999, −1.13409147605806736278049902632,
0.01065779687532857165666677581, 2.08100580542859812053489934032, 2.91432149632903365904808842496, 3.41439945873293673447109025019, 5.02920525245579307886407570957, 5.66075077209451927788831289246, 6.61487760806960272007718578294, 7.17462984931863006573743416423, 8.155014405277547345576513601136, 8.745454623888842851620621070194
