L(s) = 1 | + (0.809 + 0.587i)2-s + (0.104 + 0.994i)3-s + (0.309 + 0.951i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (2.15 + 0.458i)7-s + (−0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.104 + 0.994i)10-s + (−2.16 + 2.40i)11-s + (−0.913 + 0.406i)12-s + (−2.32 − 1.03i)13-s + (1.47 + 1.63i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (2.97 + 3.30i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (0.0603 + 0.574i)3-s + (0.154 + 0.475i)4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s + (0.815 + 0.173i)7-s + (−0.109 + 0.336i)8-s + (−0.326 + 0.0693i)9-s + (−0.0330 + 0.314i)10-s + (−0.653 + 0.726i)11-s + (−0.263 + 0.117i)12-s + (−0.644 − 0.286i)13-s + (0.394 + 0.437i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.722 + 0.802i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02506 + 2.00668i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02506 + 2.00668i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (5.53 - 0.574i)T \) |
good | 7 | \( 1 + (-2.15 - 0.458i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (2.16 - 2.40i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (2.32 + 1.03i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-2.97 - 3.30i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.74 + 1.22i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.889 - 2.73i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.29 - 1.66i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (0.584 - 1.01i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.671 + 6.39i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-5.34 + 2.37i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-1.23 + 0.897i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-6.72 + 1.42i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.699 - 6.65i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 + (4.73 + 8.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.24 - 0.902i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (2.15 - 2.38i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-9.54 - 10.6i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.568 - 5.41i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-1.97 - 6.07i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.16 + 9.74i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.38140969029064614518146212113, −9.623119650317157280460260682199, −8.568402378475718654868478707412, −7.67535145854770563489847400513, −7.09440935257506733427076276480, −5.66319306542260719638607463164, −5.25426081717718181569651674228, −4.25047237709556490576099002229, −3.16181215018206568423049487073, −2.02658831904116490947487089397,
0.893725832885628122692761237192, 2.16009697543651664081665400209, 3.19171782527328960843088426955, 4.55944248219203374796517821926, 5.30885879680421934767717623814, 6.10090322339497877678881446704, 7.35916511938005582602501877299, 7.937209442218173794210216598729, 8.965197826479122813344179313651, 9.878498174026535461232505381872