L(s) = 1 | + (−0.965 + 0.258i)3-s + (−0.707 + 0.707i)5-s + (−0.866 − 0.5i)7-s + (0.866 − 0.499i)9-s + (0.258 − 0.448i)11-s + (0.500 − 0.866i)15-s + (0.965 + 0.258i)21-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.93i·29-s + (0.866 + 0.5i)31-s + (−0.133 + 0.5i)33-s + (0.965 − 0.258i)35-s + (−0.258 + 0.965i)45-s + (0.499 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)3-s + (−0.707 + 0.707i)5-s + (−0.866 − 0.5i)7-s + (0.866 − 0.499i)9-s + (0.258 − 0.448i)11-s + (0.500 − 0.866i)15-s + (0.965 + 0.258i)21-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.93i·29-s + (0.866 + 0.5i)31-s + (−0.133 + 0.5i)33-s + (0.965 − 0.258i)35-s + (−0.258 + 0.965i)45-s + (0.499 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4467213663\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4467213663\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 11 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 - 1.93iT - T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
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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
−9.134875257785375494787424411300, −8.239437572300359479544621389036, −7.21364924917035741405843227585, −6.80851321264198808778195651006, −6.19387944113982140481813593969, −5.27692536098683843237016892392, −4.33277634845689654792838844941, −3.61226466163000569988799377879, −2.91537750021228595226378109068, −1.13623589518279083939286337176,
0.35356353233888616892835095014, 1.71607228453058383540700425256, 2.97466540525549432201677106439, 4.20385164867053708728434187058, 4.61130561425247732977945220978, 5.72041047761652358802756482290, 6.16692535428971899340427280864, 7.05020036927653153307132556538, 7.74816442080458353863600771353, 8.476206640491775599254393116345