| L(s) = 1 | + (−0.939 − 0.342i)2-s + (−1.73 + 0.0833i)3-s + (0.766 + 0.642i)4-s + (0.111 − 0.629i)5-s + (1.65 + 0.513i)6-s + (−0.766 + 0.642i)7-s + (−0.500 − 0.866i)8-s + (2.98 − 0.288i)9-s + (−0.319 + 0.553i)10-s + (0.0504 + 0.286i)11-s + (−1.37 − 1.04i)12-s + (−4.81 + 1.75i)13-s + (0.939 − 0.342i)14-s + (−0.139 + 1.09i)15-s + (0.173 + 0.984i)16-s + (2.78 − 4.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.998 + 0.0481i)3-s + (0.383 + 0.321i)4-s + (0.0496 − 0.281i)5-s + (0.675 + 0.209i)6-s + (−0.289 + 0.242i)7-s + (−0.176 − 0.306i)8-s + (0.995 − 0.0961i)9-s + (−0.101 + 0.175i)10-s + (0.0152 + 0.0862i)11-s + (−0.398 − 0.302i)12-s + (−1.33 + 0.486i)13-s + (0.251 − 0.0914i)14-s + (−0.0360 + 0.283i)15-s + (0.0434 + 0.246i)16-s + (0.676 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.625012 + 0.176896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.625012 + 0.176896i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (1.73 - 0.0833i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| good | 5 | \( 1 + (-0.111 + 0.629i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.0504 - 0.286i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (4.81 - 1.75i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.78 + 4.82i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.87 - 6.71i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.72 - 3.12i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.43 - 1.61i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.53 - 4.64i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.949 + 1.64i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.68 - 2.79i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.947 - 5.37i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.88 + 4.10i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 8.29T + 53T^{2} \) |
| 59 | \( 1 + (1.86 - 10.5i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.92 + 3.29i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (7.99 - 2.90i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.99 + 3.46i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.00 + 1.74i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.76 + 3.55i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (9.78 + 3.56i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (8.78 + 15.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.49 - 14.1i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.71630497988240693701848553352, −10.26013462059533615315391307637, −9.876506118125002310089959041569, −8.923943082227586533462097347969, −7.52022462647989584522917889046, −6.90786318126285289935973634821, −5.58518118174448025521442923013, −4.73446554000394037466465366683, −3.04422688392116672675078863323, −1.21725097752174813599695414508,
0.73342619521758851169150738651, 2.75421156188430795729490809205, 4.61042830427344790308427068661, 5.60222268464051456084264938159, 6.73504972996304359072311297434, 7.27166295686422025112787334199, 8.460455138491702246822909704416, 9.775869405901879677298499800663, 10.27318132902574916789312757949, 11.10092287024071829441550318324
