| L(s) = 1 | − 1.73i·2-s − 0.999·4-s + 2i·7-s − 1.73i·8-s + 1.26·11-s − i·13-s + 3.46·14-s − 5·16-s − 3.46i·17-s − 4.19·19-s − 2.19i·22-s + 4.73i·23-s − 1.73·26-s − 1.99i·28-s − 9.46·29-s + ⋯ |
| L(s) = 1 | − 1.22i·2-s − 0.499·4-s + 0.755i·7-s − 0.612i·8-s + 0.382·11-s − 0.277i·13-s + 0.925·14-s − 1.25·16-s − 0.840i·17-s − 0.962·19-s − 0.468i·22-s + 0.986i·23-s − 0.339·26-s − 0.377i·28-s − 1.75·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9990051270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9990051270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 - 4.73iT - 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 0.196T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 14.3iT - 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.682067603412160303138193582795, −7.55319015170431360818489088737, −6.87890540383959885100158069957, −5.88864548771994632079752246752, −5.16013552318825405850929308877, −4.02168855076443168227382526579, −3.40160356120677990070321037684, −2.36861072063877054354545862985, −1.74959656484480713477710329618, −0.29592838283879297053426078111,
1.48854999134225798043253932219, 2.67753998875603211603779884847, 4.07836112344804963563079264050, 4.48432952235178979493882154750, 5.67474730590521190985551848732, 6.25928306767271212202268535118, 6.89139366922924470404000224441, 7.59521004914697949587847058007, 8.235198791927701891295450585958, 8.929782445542850969099263129152
