| L(s) = 1 | + 3-s + 1.41·5-s + 9-s + 2.82·11-s + 1.41·13-s + 1.41·15-s − 1.41·17-s − 2.82·23-s − 2.99·25-s + 27-s + 4·31-s + 2.82·33-s + 4·37-s + 1.41·39-s + 1.41·41-s + 5.65·43-s + 1.41·45-s + 12·47-s − 1.41·51-s + 10·53-s + 4.00·55-s + 1.41·61-s + 2.00·65-s + 11.3·67-s − 2.82·69-s + 2.82·71-s − 12.7·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.632·5-s + 0.333·9-s + 0.852·11-s + 0.392·13-s + 0.365·15-s − 0.342·17-s − 0.589·23-s − 0.599·25-s + 0.192·27-s + 0.718·31-s + 0.492·33-s + 0.657·37-s + 0.226·39-s + 0.220·41-s + 0.862·43-s + 0.210·45-s + 1.75·47-s − 0.198·51-s + 1.37·53-s + 0.539·55-s + 0.181·61-s + 0.248·65-s + 1.38·67-s − 0.340·69-s + 0.335·71-s − 1.48·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.056278217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.056278217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−8.420358360920155440011965508138, −7.60454192892165111312237689324, −6.85716807234475903180502497356, −6.10386583191583754360314221033, −5.54012557139512352883733429909, −4.30815073603487917390288963355, −3.89922123707437423750996842736, −2.74792362411384573287536640634, −2.01526402336484699714526873711, −0.988085696305700639181721761246,
0.988085696305700639181721761246, 2.01526402336484699714526873711, 2.74792362411384573287536640634, 3.89922123707437423750996842736, 4.30815073603487917390288963355, 5.54012557139512352883733429909, 6.10386583191583754360314221033, 6.85716807234475903180502497356, 7.60454192892165111312237689324, 8.420358360920155440011965508138
