L(s) = 1 | + (2.72 − 4.72i)2-s + (−10.8 − 18.8i)4-s + (2.5 + 4.33i)5-s + (5.90 − 10.2i)7-s − 75.3·8-s + 27.2·10-s + (−28.1 + 48.7i)11-s + (−17.2 − 29.9i)13-s + (−32.2 − 55.8i)14-s + (−118. + 205. i)16-s − 39.2·17-s − 146.·19-s + (54.4 − 94.3i)20-s + (153. + 265. i)22-s + (−11.7 − 20.4i)23-s + ⋯ |
L(s) = 1 | + (0.964 − 1.67i)2-s + (−1.36 − 2.35i)4-s + (0.223 + 0.387i)5-s + (0.318 − 0.552i)7-s − 3.32·8-s + 0.863·10-s + (−0.770 + 1.33i)11-s + (−0.369 − 0.639i)13-s + (−0.615 − 1.06i)14-s + (−1.84 + 3.20i)16-s − 0.560·17-s − 1.76·19-s + (0.609 − 1.05i)20-s + (1.48 + 2.57i)22-s + (−0.106 − 0.185i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9142265283\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9142265283\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 2 | \( 1 + (-2.72 + 4.72i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-5.90 + 10.2i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (28.1 - 48.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (17.2 + 29.9i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 39.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (11.7 + 20.4i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-80.5 + 139. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-14.7 - 25.5i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-71.1 - 123. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-234. + 405. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-197. + 341. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 134.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (65.5 + 113. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (129. - 224. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (222. + 385. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 560.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + (225. - 390. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-142. + 246. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 625.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-96.6 + 167. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.46650600631741181612864536676, −9.854058931291739894693732248522, −8.606736101515439971069034128603, −7.12459093286547223426266193428, −5.85187980729191149239131415119, −4.72717062766255060109147330791, −4.10573871059460405852811956766, −2.63838422845495727945446969161, −1.92925186667533687559979927281, −0.20286553645390218391544323662,
2.66356653340541148682512127676, 4.10824151821342773621331874258, 4.99987031855558922792452249623, 5.86040924338431838671865863765, 6.57078449277301893363258385739, 7.74199155865062614443142501211, 8.627527968799142941498323354537, 8.998732373358810126682504628944, 10.75957337016333461762870011356, 11.93863787614944708461176707457