L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.707 − 1.22i)19-s + (−0.5 − 0.866i)25-s + (−0.499 − 0.866i)32-s + 1.41·34-s + (0.707 + 1.22i)38-s − 1.41·41-s + 0.999·50-s + (−0.707 − 1.22i)59-s + 0.999·64-s + (1 + 1.73i)67-s + (−0.707 + 1.22i)68-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.707 − 1.22i)19-s + (−0.5 − 0.866i)25-s + (−0.499 − 0.866i)32-s + 1.41·34-s + (0.707 + 1.22i)38-s − 1.41·41-s + 0.999·50-s + (−0.707 − 1.22i)59-s + 0.999·64-s + (1 + 1.73i)67-s + (−0.707 + 1.22i)68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7396200125\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7396200125\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−8.635271897428709529382489497989, −7.938948155557544694554147825907, −7.07479430306699047316323794013, −6.72462101648917847911808398627, −5.75633845321950685573796082268, −4.96070999035228669709231458711, −4.41086108454520350672420838128, −3.10589249017634638689089847983, −1.98523321486003270273318274431, −0.53661693225407695761098432685,
1.37156773131665540528566931529, 2.14987089495500068427035132349, 3.37248104038385488029820420339, 3.87247669782138582780345235465, 4.86681238347315266997285581112, 5.77763032657000889302325771798, 6.71157398513552065834838634133, 7.63831594265698707798775227941, 8.176245117371032844428910751283, 8.896443951494270257944402578214