| L(s) = 1 | + (−1.99 − 1.00i)5-s − 0.0883·7-s + (−0.701 + 2.15i)11-s + (0.819 + 2.52i)13-s + (1.68 − 1.22i)17-s + (−1.42 + 1.03i)19-s + (−1.46 + 4.50i)23-s + (2.98 + 4.00i)25-s + (2.99 + 2.17i)29-s + (3.32 − 2.41i)31-s + (0.176 + 0.0885i)35-s + (2.19 + 6.77i)37-s + (2.03 + 6.26i)41-s + 1.79·43-s + (8.17 + 5.94i)47-s + ⋯ |
| L(s) = 1 | + (−0.893 − 0.448i)5-s − 0.0333·7-s + (−0.211 + 0.650i)11-s + (0.227 + 0.699i)13-s + (0.409 − 0.297i)17-s + (−0.326 + 0.237i)19-s + (−0.305 + 0.940i)23-s + (0.597 + 0.801i)25-s + (0.556 + 0.404i)29-s + (0.596 − 0.433i)31-s + (0.0298 + 0.0149i)35-s + (0.361 + 1.11i)37-s + (0.318 + 0.978i)41-s + 0.273·43-s + (1.19 + 0.866i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.871730 + 0.600237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.871730 + 0.600237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.99 + 1.00i)T \) |
| good | 7 | \( 1 + 0.0883T + 7T^{2} \) |
| 11 | \( 1 + (0.701 - 2.15i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.819 - 2.52i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.68 + 1.22i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.42 - 1.03i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.46 - 4.50i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.99 - 2.17i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.32 + 2.41i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.19 - 6.77i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.03 - 6.26i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 1.79T + 43T^{2} \) |
| 47 | \( 1 + (-8.17 - 5.94i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.777 + 0.565i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.77 - 8.53i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.92 + 9.00i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (11.2 - 8.17i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-4.97 - 3.61i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.975 - 3.00i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (10.8 + 7.84i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (12.7 - 9.29i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.16 + 9.74i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-10.3 - 7.52i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.17112220082594831349847662179, −9.442735492624062007974140130984, −8.515176304207333273248909831030, −7.76860988263069563101186377123, −7.01641303106685338274545458750, −5.93585723614689124541679886727, −4.76660743582283229802807542215, −4.11819558748849824820631247471, −2.91713910670817933112151553192, −1.34158935957829744362915568505,
0.55579295499197430820262194501, 2.55764647172684124755688855802, 3.52617602164327337925410260610, 4.45203475594944412494884165057, 5.66944794990221622854271893574, 6.52619462134555450537689913719, 7.52028641781462585629693242715, 8.233725971027205163948039435745, 8.900204602988919811604380253413, 10.27154636665216212329537828773
