| L(s) = 1 | + (2.81 − 2.60i)2-s + (5.69 − 0.858i)3-s + (0.501 − 6.69i)4-s + (−1.18 + 3.01i)5-s + (13.7 − 17.2i)6-s + (−17.2 + 6.81i)7-s + (3.07 + 3.85i)8-s + (5.91 − 1.82i)9-s + (4.54 + 11.5i)10-s + (−25.5 − 7.87i)11-s + (−2.89 − 38.5i)12-s + (−7.17 − 31.4i)13-s + (−30.6 + 64.1i)14-s + (−4.15 + 18.2i)15-s + (71.8 + 10.8i)16-s + (18.4 − 12.5i)17-s + ⋯ |
| L(s) = 1 | + (0.994 − 0.922i)2-s + (1.09 − 0.165i)3-s + (0.0627 − 0.836i)4-s + (−0.105 + 0.269i)5-s + (0.937 − 1.17i)6-s + (−0.929 + 0.368i)7-s + (0.135 + 0.170i)8-s + (0.218 − 0.0675i)9-s + (0.143 + 0.366i)10-s + (−0.700 − 0.215i)11-s + (−0.0695 − 0.927i)12-s + (−0.153 − 0.670i)13-s + (−0.584 + 1.22i)14-s + (−0.0715 + 0.313i)15-s + (1.12 + 0.169i)16-s + (0.263 − 0.179i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.27429 - 1.24011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.27429 - 1.24011i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (17.2 - 6.81i)T \) |
| good | 2 | \( 1 + (-2.81 + 2.60i)T + (0.597 - 7.97i)T^{2} \) |
| 3 | \( 1 + (-5.69 + 0.858i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (1.18 - 3.01i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (25.5 + 7.87i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (7.17 + 31.4i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (-18.4 + 12.5i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (22.4 + 38.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-134. - 91.7i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (107. + 51.8i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-71.5 + 123. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-10.6 - 141. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (217. + 273. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-185. + 232. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (107. - 99.3i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (-16.4 + 219. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (132. + 338. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (-50.0 - 667. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (544. - 943. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (438. - 211. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (702. + 651. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (-485. - 841. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-11.9 + 52.5i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-1.12e3 + 347. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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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
−14.69907567786589419148166632546, −13.29019421574914648117820292870, −13.13202931124232472754131278461, −11.65056893967786739648899561716, −10.38502139642137664666318605082, −8.949769892068529076275794825765, −7.51468410341745431173712375964, −5.43293762967896725352007572621, −3.40023443089192918063842167959, −2.62420566207152579711005252320,
3.20641662457481825666072576033, 4.63976063074498879360315319277, 6.37529776007523750778359011339, 7.63483233306616876489657460060, 9.008932057282071983590172765219, 10.33603627329899135541141947170, 12.53166642597877920626604671084, 13.31073896261983341598438525412, 14.32475012209275078441885596882, 15.02700125580127592901733653046
