L(s) = 1 | + (−1.88 − 7.03i)2-s + (11.2 − 6.51i)3-s + (−32.0 + 18.4i)4-s + (9.58 − 35.7i)5-s + (−67.0 − 67.0i)6-s + (25.6 + 44.4i)7-s + (108. + 108. i)8-s + (44.3 − 76.7i)9-s − 269.·10-s + 39.1i·11-s + (−240. + 417. i)12-s + (−60.4 + 225. i)13-s + (264. − 264. i)14-s + (−124. − 466. i)15-s + (260. − 450. i)16-s + (318. − 85.3i)17-s + ⋯ |
L(s) = 1 | + (−0.471 − 1.75i)2-s + (1.25 − 0.723i)3-s + (−2.00 + 1.15i)4-s + (0.383 − 1.43i)5-s + (−1.86 − 1.86i)6-s + (0.524 + 0.907i)7-s + (1.68 + 1.68i)8-s + (0.547 − 0.948i)9-s − 2.69·10-s + 0.323i·11-s + (−1.67 + 2.89i)12-s + (−0.357 + 1.33i)13-s + (1.34 − 1.34i)14-s + (−0.554 − 2.07i)15-s + (1.01 − 1.76i)16-s + (1.10 − 0.295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0961159 - 1.59221i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0961159 - 1.59221i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-1.27e3 - 496. i)T \) |
good | 2 | \( 1 + (1.88 + 7.03i)T + (-13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (-11.2 + 6.51i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-9.58 + 35.7i)T + (-541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-25.6 - 44.4i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 39.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (60.4 - 225. i)T + (-2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-318. + 85.3i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-89.0 + 332. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (581. + 581. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (-109. + 109. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (180. - 180. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (1.73e3 - 1.00e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.51e3 - 1.51e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 761.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (1.10e3 - 1.92e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.47e3 - 394. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-3.17e3 - 851. i)T + (1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (3.77e3 - 2.17e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-4.04 - 7.00i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + 6.85e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (1.43e3 - 5.36e3i)T + (-3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-843. + 1.46e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-665. - 2.48e3i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (751. + 751. i)T + 8.85e7iT^{2} \) |
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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
−14.42639307059308220934435143332, −13.42772077922312701564554436587, −12.44667569545315843164383414591, −11.79244063203185560222222515677, −9.625050993525072407736219759895, −8.951922584915276787632285206329, −8.101526658841427496783848704935, −4.63363735285810043019566476764, −2.48074684991110360603838617251, −1.42483191043945117693933170971,
3.59629718605467793999398036927, 5.74582710723056329996878039744, 7.48669649409013186307937387762, 8.073167954189998470852171855871, 9.802156860732314059027542575328, 10.38091311544407799017518699194, 13.73801847400745803203020271720, 14.36239785673187440247306149511, 14.89328561149259057880897551873, 15.88810905019027130725753391911