L(s) = 1 | + 1.99i·2-s − 1.58·3-s − 1.97·4-s + 2.66i·5-s − 3.16i·6-s − 1.73i·7-s + 0.0439i·8-s − 0.476·9-s − 5.32·10-s + 2.93i·11-s + 3.14·12-s + (−3.42 − 1.12i)13-s + 3.46·14-s − 4.23i·15-s − 4.04·16-s − 1.90·17-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 0.917·3-s − 0.988·4-s + 1.19i·5-s − 1.29i·6-s − 0.657i·7-s + 0.0155i·8-s − 0.158·9-s − 1.68·10-s + 0.886i·11-s + 0.906·12-s + (−0.949 − 0.312i)13-s + 0.927·14-s − 1.09i·15-s − 1.01·16-s − 0.461·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.266370 - 0.368015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.266370 - 0.368015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (3.42 + 1.12i)T \) |
| 31 | \( 1 - iT \) |
good | 2 | \( 1 - 1.99iT - 2T^{2} \) |
| 3 | \( 1 + 1.58T + 3T^{2} \) |
| 5 | \( 1 - 2.66iT - 5T^{2} \) |
| 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 2.93iT - 11T^{2} \) |
| 17 | \( 1 + 1.90T + 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 37 | \( 1 - 3.05iT - 37T^{2} \) |
| 41 | \( 1 + 5.80iT - 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 13.7iT - 59T^{2} \) |
| 61 | \( 1 - 5.05T + 61T^{2} \) |
| 67 | \( 1 - 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 5.83iT - 71T^{2} \) |
| 73 | \( 1 + 4.92iT - 73T^{2} \) |
| 79 | \( 1 - 0.743T + 79T^{2} \) |
| 83 | \( 1 + 0.165iT - 83T^{2} \) |
| 89 | \( 1 - 1.85iT - 89T^{2} \) |
| 97 | \( 1 - 6.35iT - 97T^{2} \) |
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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
−11.65831476272166502655644411169, −10.97627537415658053162310905441, −10.19666222417065809329737636702, −9.019907685991616334129645787671, −7.65940193607962270678157146362, −6.98434077165417412341121242272, −6.56223410730717966446801649104, −5.37405724872223766726919824907, −4.57731353223789115205886834875, −2.72346539437469374169943870977,
0.31724277709481984017687016418, 1.83376102809879966433373630506, 3.29769303978150257956464635104, 4.72100358947154088691213871683, 5.42073066789857775087095745676, 6.57176890388937031872584393033, 8.270540120756451826735969149257, 9.084616149914109913593540343141, 9.860012887954088825720643986430, 11.00038108242604399547689877696