| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s − 4.89i·7-s + i·8-s − 9-s − i·12-s − 6.89i·13-s − 4.89·14-s + 16-s − i·17-s + i·18-s − 4·19-s + 4.89·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.85i·7-s + 0.353i·8-s − 0.333·9-s − 0.288i·12-s − 1.91i·13-s − 1.30·14-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s − 0.917·19-s + 1.06·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6827306451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6827306451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 + 4.89iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.89iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 + 8.89iT - 43T^{2} \) |
| 47 | \( 1 - 9.79iT - 47T^{2} \) |
| 53 | \( 1 - 7.79iT - 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 0.898iT - 67T^{2} \) |
| 71 | \( 1 + 8.89T + 71T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 5.79T + 79T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 - 7.79T + 89T^{2} \) |
| 97 | \( 1 + 12.6iT - 97T^{2} \) |
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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
−8.453951603963776004000052933803, −7.80839877400707367465510149997, −7.11793257374426857737511073084, −5.96059103336306500130932082861, −5.08516545346670090866997663480, −4.25690944728786565723072602545, −3.59533071939323394580718895463, −2.84309354136588523945401882728, −1.29622532570298227125708074685, −0.23029475477837846189140319816,
1.83710965086485741568466836473, 2.44875501827474828794991139577, 3.85383124520075849115852723334, 4.83625050402466684097425728755, 5.62657916734905469164180277613, 6.48377644893492291134623305117, 6.69982464492988550640831372878, 7.924492857584280856433347010018, 8.571006849963097735740886955807, 9.072065136627150708703443962336
