L(s) = 1 | − 1.36·2-s + 3-s − 0.128·4-s + 2.18·5-s − 1.36·6-s − 4.27·7-s + 2.91·8-s + 9-s − 2.98·10-s + 11-s − 0.128·12-s + 5.84·14-s + 2.18·15-s − 3.72·16-s − 3.79·17-s − 1.36·18-s − 3.17·19-s − 0.279·20-s − 4.27·21-s − 1.36·22-s − 4.94·23-s + 2.91·24-s − 0.231·25-s + 27-s + 0.547·28-s − 7.35·29-s − 2.98·30-s + ⋯ |
L(s) = 1 | − 0.967·2-s + 0.577·3-s − 0.0640·4-s + 0.976·5-s − 0.558·6-s − 1.61·7-s + 1.02·8-s + 0.333·9-s − 0.944·10-s + 0.301·11-s − 0.0370·12-s + 1.56·14-s + 0.563·15-s − 0.931·16-s − 0.919·17-s − 0.322·18-s − 0.727·19-s − 0.0625·20-s − 0.931·21-s − 0.291·22-s − 1.03·23-s + 0.594·24-s − 0.0463·25-s + 0.192·27-s + 0.103·28-s − 1.36·29-s − 0.545·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9463976798\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9463976798\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 + 4.27T + 7T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 3.17T + 19T^{2} \) |
| 23 | \( 1 + 4.94T + 23T^{2} \) |
| 29 | \( 1 + 7.35T + 29T^{2} \) |
| 31 | \( 1 - 0.0727T + 31T^{2} \) |
| 37 | \( 1 - 3.66T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 5.11T + 53T^{2} \) |
| 59 | \( 1 - 5.62T + 59T^{2} \) |
| 61 | \( 1 + 5.40T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 9.62T + 71T^{2} \) |
| 73 | \( 1 - 6.44T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.80T + 83T^{2} \) |
| 89 | \( 1 - 2.87T + 89T^{2} \) |
| 97 | \( 1 - 3.79T + 97T^{2} \) |
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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
−8.377215501427520152125672402504, −7.52539544843378752501396396281, −6.80051321990990396812518796733, −6.20530333182240284917056737850, −5.46627730094017208196917672580, −4.18910870028422016206512592871, −3.73716937714775075868607038537, −2.44519818687023432966200326740, −1.95411892567305961799573931540, −0.56615957654281810379048850305,
0.56615957654281810379048850305, 1.95411892567305961799573931540, 2.44519818687023432966200326740, 3.73716937714775075868607038537, 4.18910870028422016206512592871, 5.46627730094017208196917672580, 6.20530333182240284917056737850, 6.80051321990990396812518796733, 7.52539544843378752501396396281, 8.377215501427520152125672402504