| L(s) = 1 | + 1.12i·3-s + (0.981 + 2.00i)5-s + 1.74·9-s − 4.94·11-s − 4.64i·13-s + (−2.25 + 1.10i)15-s − 5.72i·17-s − 7.53·19-s + 6.87i·23-s + (−3.07 + 3.94i)25-s + 5.31i·27-s − 8.11·29-s − 3.87·31-s − 5.54i·33-s + 5.18i·37-s + ⋯ |
| L(s) = 1 | + 0.647i·3-s + (0.438 + 0.898i)5-s + 0.580·9-s − 1.49·11-s − 1.28i·13-s + (−0.582 + 0.284i)15-s − 1.38i·17-s − 1.72·19-s + 1.43i·23-s + (−0.615 + 0.788i)25-s + 1.02i·27-s − 1.50·29-s − 0.696·31-s − 0.965i·33-s + 0.852i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3561062336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3561062336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.981 - 2.00i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.12iT - 3T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 + 4.64iT - 13T^{2} \) |
| 17 | \( 1 + 5.72iT - 17T^{2} \) |
| 19 | \( 1 + 7.53T + 19T^{2} \) |
| 23 | \( 1 - 6.87iT - 23T^{2} \) |
| 29 | \( 1 + 8.11T + 29T^{2} \) |
| 31 | \( 1 + 3.87T + 31T^{2} \) |
| 37 | \( 1 - 5.18iT - 37T^{2} \) |
| 41 | \( 1 + 9.45T + 41T^{2} \) |
| 43 | \( 1 - 0.706iT - 43T^{2} \) |
| 47 | \( 1 - 2.20iT - 47T^{2} \) |
| 53 | \( 1 - 1.32iT - 53T^{2} \) |
| 59 | \( 1 - 3.94T + 59T^{2} \) |
| 61 | \( 1 + 6.07T + 61T^{2} \) |
| 67 | \( 1 + 8.24iT - 67T^{2} \) |
| 71 | \( 1 + 4.50T + 71T^{2} \) |
| 73 | \( 1 + 2.93iT - 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 5.27iT - 83T^{2} \) |
| 89 | \( 1 - 9.49T + 89T^{2} \) |
| 97 | \( 1 + 1.74iT - 97T^{2} \) |
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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
−9.775259063263681883499696948305, −9.070955615766099755984466078531, −7.81122742735510741515720928013, −7.46242302199920745014718755148, −6.45985869104573864303918607333, −5.41273957886355904882638450032, −5.00592848604343243407998135850, −3.67986967207717593003041054132, −2.96654203641292287810972538491, −1.95550372302788511188688287202,
0.11370467947373022312143952009, 1.84106288303567026494593768895, 2.13085951942728824474927162904, 3.97181726889178491547525207949, 4.58597693729523400574179279842, 5.60605094465687172750546963149, 6.38206667330671198041670891252, 7.11674122679908040196404273625, 8.126265332053888189787481666981, 8.563008659163099730418397855448
