L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (−0.5 − 0.866i)6-s + (−0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.766 + 1.32i)11-s + (0.173 + 0.984i)12-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (0.173 − 0.984i)18-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (0.173 − 0.984i)24-s + (0.766 + 0.642i)25-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (−0.5 − 0.866i)6-s + (−0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.766 + 1.32i)11-s + (0.173 + 0.984i)12-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (0.173 − 0.984i)18-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (0.173 − 0.984i)24-s + (0.766 + 0.642i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8087379840\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8087379840\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (0.939 - 0.342i)T \) |
good | 5 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 - 0.347T + T^{2} \) |
| 89 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
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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
−10.03207380736555093772018079819, −8.948046567713926529798551226215, −8.769636059091292721164812095272, −7.56736019353816724831494730967, −7.24502468041259360662276318661, −5.91742472623838857256860854639, −4.59808224015207376683219126202, −3.85014262358980424989852617003, −2.58776348699621325958003745138, −1.93817065188632081708200336201,
0.816541839630491260965869587170, 2.37030091056624206715985349479, 2.98005393032754665986998964040, 4.55211341208655414162086142303, 5.92124389340760830684436596547, 6.45791984511241320261166123584, 7.43331569231267615829307100097, 8.033505694401821699922406230046, 8.836920787296374297964417210058, 9.169333400745475761564380857726