| L(s) = 1 | + (0.904 − 1.56i)2-s + (1.99 + 4.79i)3-s + (2.36 + 4.09i)4-s − 2.09·5-s + (9.32 + 1.21i)6-s + (−11.3 + 14.6i)7-s + 23.0·8-s + (−19.0 + 19.1i)9-s + (−1.89 + 3.28i)10-s + 23.4·11-s + (−14.9 + 19.5i)12-s + (27.8 − 48.2i)13-s + (12.7 + 30.9i)14-s + (−4.18 − 10.0i)15-s + (1.91 − 3.32i)16-s + (55.3 − 95.9i)17-s + ⋯ |
| L(s) = 1 | + (0.319 − 0.553i)2-s + (0.383 + 0.923i)3-s + (0.295 + 0.511i)4-s − 0.187·5-s + (0.634 + 0.0826i)6-s + (−0.611 + 0.791i)7-s + 1.01·8-s + (−0.705 + 0.709i)9-s + (−0.0600 + 0.103i)10-s + 0.643·11-s + (−0.359 + 0.469i)12-s + (0.594 − 1.02i)13-s + (0.242 + 0.591i)14-s + (−0.0720 − 0.173i)15-s + (0.0299 − 0.0519i)16-s + (0.790 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.78313 + 0.717617i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78313 + 0.717617i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.99 - 4.79i)T \) |
| 7 | \( 1 + (11.3 - 14.6i)T \) |
| good | 2 | \( 1 + (-0.904 + 1.56i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + 2.09T + 125T^{2} \) |
| 11 | \( 1 - 23.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-27.8 + 48.2i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-55.3 + 95.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-9.75 - 16.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 13.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-59.9 - 103. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (79.4 + 137. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (79.4 + 137. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-208. + 361. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-131. - 227. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (95.1 - 164. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-175. + 304. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-63.9 - 110. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (362. - 627. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-499. - 865. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 404.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (120. - 208. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-460. + 798. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (502. + 869. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (239. + 414. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-431. - 747. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.59268578298252741345165464946, −13.41368007846860073033201131631, −12.22398614085320001723572905169, −11.33075945321654157736244521807, −10.08659326724472021311693185272, −8.929604719355903749130866964352, −7.61403135473019555285410623533, −5.59514735995576666878945229032, −3.84114460609760330359247340551, −2.79024066454715391753181833599,
1.43959410260062879920912736212, 3.88225989965481041001066651468, 6.10165978959898971451897749093, 6.83273173978756133915144947112, 8.032256096149853272693255084439, 9.621516175597234187355796957045, 11.02888298196021914777922783550, 12.30552676127182994595980177788, 13.61086267300181319808154025694, 14.14633099966048941218683428766
