| L(s) = 1 | − 3.20i·3-s + (2.02 − 0.945i)5-s + 4.54i·7-s − 7.29·9-s − 4.59·11-s − 5.17i·13-s + (−3.03 − 6.50i)15-s + 3.33i·17-s − 4.43·19-s + 14.5·21-s + i·23-s + (3.21 − 3.83i)25-s + 13.8i·27-s − 4.13·29-s − 7.82·31-s + ⋯ |
| L(s) = 1 | − 1.85i·3-s + (0.906 − 0.423i)5-s + 1.71i·7-s − 2.43·9-s − 1.38·11-s − 1.43i·13-s + (−0.783 − 1.67i)15-s + 0.809i·17-s − 1.01·19-s + 3.18·21-s + 0.208i·23-s + (0.642 − 0.766i)25-s + 2.65i·27-s − 0.766·29-s − 1.40·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.423 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2501864348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2501864348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.02 + 0.945i)T \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 3.20iT - 3T^{2} \) |
| 7 | \( 1 - 4.54iT - 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 5.17iT - 13T^{2} \) |
| 17 | \( 1 - 3.33iT - 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 + 7.82T + 31T^{2} \) |
| 37 | \( 1 + 1.03iT - 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 - 1.67iT - 43T^{2} \) |
| 47 | \( 1 + 6.20iT - 47T^{2} \) |
| 53 | \( 1 - 7.35iT - 53T^{2} \) |
| 59 | \( 1 - 1.83T + 59T^{2} \) |
| 61 | \( 1 - 0.524T + 61T^{2} \) |
| 67 | \( 1 - 2.55iT - 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 - 2.41iT - 73T^{2} \) |
| 79 | \( 1 + 9.85T + 79T^{2} \) |
| 83 | \( 1 + 10.7iT - 83T^{2} \) |
| 89 | \( 1 + 5.13T + 89T^{2} \) |
| 97 | \( 1 + 3.36iT - 97T^{2} \) |
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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
−8.404848139559496536759930373313, −8.147394960790228325938100189542, −7.15929079507496805843891696056, −6.09268780026426358983593958423, −5.64475955151112241787564013817, −5.27231138638300260070200779499, −3.03671722980215277601993217290, −2.30170960475056388607658095480, −1.72235620373539224898592083077, −0.080688884815721004191447698121,
2.13563420551671027901589024434, 3.27933002202998494279523652141, 4.08135979366559450669847582512, 4.78921737479619832313247774219, 5.48050758433393733796074992981, 6.58958312542455602082410713877, 7.34502378704717521107887288481, 8.456741931982538767728639226455, 9.388091391231296027605031383950, 9.807642928717241389104199624872
