| L(s) = 1 | − 0.112·2-s + 3-s − 1.98·4-s + 1.06·5-s − 0.112·6-s − 7-s + 0.448·8-s + 9-s − 0.119·10-s − 1.98·12-s − 5.84·13-s + 0.112·14-s + 1.06·15-s + 3.92·16-s + 2.80·17-s − 0.112·18-s − 3.56·19-s − 2.10·20-s − 21-s + 4.72·23-s + 0.448·24-s − 3.87·25-s + 0.657·26-s + 27-s + 1.98·28-s + 8.59·29-s − 0.119·30-s + ⋯ |
| L(s) = 1 | − 0.0795·2-s + 0.577·3-s − 0.993·4-s + 0.474·5-s − 0.0459·6-s − 0.377·7-s + 0.158·8-s + 0.333·9-s − 0.0377·10-s − 0.573·12-s − 1.62·13-s + 0.0300·14-s + 0.274·15-s + 0.981·16-s + 0.680·17-s − 0.0265·18-s − 0.817·19-s − 0.471·20-s − 0.218·21-s + 0.984·23-s + 0.0915·24-s − 0.774·25-s + 0.128·26-s + 0.192·27-s + 0.375·28-s + 1.59·29-s − 0.0217·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.587741775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587741775\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.112T + 2T^{2} \) |
| 5 | \( 1 - 1.06T + 5T^{2} \) |
| 13 | \( 1 + 5.84T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 - 1.74T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 - 6.13T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 4.20T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 - 1.16T + 83T^{2} \) |
| 89 | \( 1 - 2.90T + 89T^{2} \) |
| 97 | \( 1 + 4.85T + 97T^{2} \) |
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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
−9.007530982039689584308625108134, −8.201866192873222667069235146763, −7.56408491880633134979711632030, −6.66469126497137778358232430751, −5.69170758367459358357294125427, −4.84569211865511474616702079715, −4.19889480866155245843671047102, −3.08666034752823772739176060290, −2.27977473963659989488322437907, −0.78817511761074487140637044959,
0.78817511761074487140637044959, 2.27977473963659989488322437907, 3.08666034752823772739176060290, 4.19889480866155245843671047102, 4.84569211865511474616702079715, 5.69170758367459358357294125427, 6.66469126497137778358232430751, 7.56408491880633134979711632030, 8.201866192873222667069235146763, 9.007530982039689584308625108134
