L(s) = 1 | + i·2-s + (−0.536 + 0.536i)3-s − 4-s + (−0.127 − 2.23i)5-s + (−0.536 − 0.536i)6-s + (0.767 − 0.767i)7-s − i·8-s + 2.42i·9-s + (2.23 − 0.127i)10-s + 4.39i·11-s + (0.536 − 0.536i)12-s + 6.74i·13-s + (0.767 + 0.767i)14-s + (1.26 + 1.12i)15-s + 16-s + 7.34·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.309 + 0.309i)3-s − 0.5·4-s + (−0.0571 − 0.998i)5-s + (−0.219 − 0.219i)6-s + (0.290 − 0.290i)7-s − 0.353i·8-s + 0.808i·9-s + (0.705 − 0.0404i)10-s + 1.32i·11-s + (0.154 − 0.154i)12-s + 1.87i·13-s + (0.205 + 0.205i)14-s + (0.326 + 0.291i)15-s + 0.250·16-s + 1.78·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.770237 + 0.853236i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.770237 + 0.853236i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (0.127 + 2.23i)T \) |
| 37 | \( 1 + (-0.633 + 6.04i)T \) |
good | 3 | \( 1 + (0.536 - 0.536i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.767 + 0.767i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.39iT - 11T^{2} \) |
| 13 | \( 1 - 6.74iT - 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 + (-2.59 + 2.59i)T - 19iT^{2} \) |
| 23 | \( 1 - 1.20iT - 23T^{2} \) |
| 29 | \( 1 + (-1.25 - 1.25i)T + 29iT^{2} \) |
| 31 | \( 1 + (4.14 - 4.14i)T - 31iT^{2} \) |
| 41 | \( 1 - 4.07iT - 41T^{2} \) |
| 43 | \( 1 + 8.56iT - 43T^{2} \) |
| 47 | \( 1 + (-7.68 + 7.68i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.31 - 2.31i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.61 - 7.61i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.14 + 1.14i)T - 61iT^{2} \) |
| 67 | \( 1 + (6.25 + 6.25i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + (-1.88 + 1.88i)T - 73iT^{2} \) |
| 79 | \( 1 + (-5.13 + 5.13i)T - 79iT^{2} \) |
| 83 | \( 1 + (-0.570 - 0.570i)T + 83iT^{2} \) |
| 89 | \( 1 + (-7.54 - 7.54i)T + 89iT^{2} \) |
| 97 | \( 1 + 5.39T + 97T^{2} \) |
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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
−11.87775403334374526129154984534, −10.58538539327553507647788555620, −9.586734391055184843640757941294, −8.962013743652708927951473987377, −7.67551358557302611567445729102, −7.15206200398025890514915211627, −5.58868235098394332258762137875, −4.84058815952617061492181405325, −4.09530760749557039653801238308, −1.66411611642409934407879915869,
0.912822125550698145406151575044, 2.99610960525904060551216143003, 3.51313607123125065790536073523, 5.57743126617216937115142662559, 6.00781506788362678942224695605, 7.58969780413790904956695628062, 8.245143365650925993268842686868, 9.639765280023336698551627948769, 10.38841945395520523691190592148, 11.20105363658333859369729004895