| L(s) = 1 | + 1.61·3-s − 3.23·5-s + 0.236·7-s − 0.381·9-s + 4.38·11-s + 3.47·13-s − 5.23·15-s + 7.23·17-s + 0.381·21-s − 5.85·23-s + 5.47·25-s − 5.47·27-s − 2.85·29-s + 4.38·31-s + 7.09·33-s − 0.763·35-s − 6.38·37-s + 5.61·39-s + 7.94·41-s + 0.618·43-s + 1.23·45-s − 12.7·47-s − 6.94·49-s + 11.7·51-s + 3.85·53-s − 14.1·55-s + 5.38·59-s + ⋯ |
| L(s) = 1 | + 0.934·3-s − 1.44·5-s + 0.0892·7-s − 0.127·9-s + 1.32·11-s + 0.962·13-s − 1.35·15-s + 1.75·17-s + 0.0833·21-s − 1.22·23-s + 1.09·25-s − 1.05·27-s − 0.529·29-s + 0.787·31-s + 1.23·33-s − 0.129·35-s − 1.04·37-s + 0.899·39-s + 1.24·41-s + 0.0942·43-s + 0.184·45-s − 1.85·47-s − 0.992·49-s + 1.63·51-s + 0.529·53-s − 1.91·55-s + 0.700·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.306812202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.306812202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 - 4.38T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 - 4.38T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 - 0.618T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 - 5.38T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 - 6.70T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 7.76T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.279681701868830168797054766565, −7.69772393817729166402825695630, −6.82637883310086107246478595949, −6.07810986451570454695002643792, −5.16396357549855715465567228429, −3.90898778870189320354036443375, −3.78393777125075472783155626989, −3.13382494561391261233951004833, −1.85984389104259731366069536412, −0.791739815467697983454515623779,
0.791739815467697983454515623779, 1.85984389104259731366069536412, 3.13382494561391261233951004833, 3.78393777125075472783155626989, 3.90898778870189320354036443375, 5.16396357549855715465567228429, 6.07810986451570454695002643792, 6.82637883310086107246478595949, 7.69772393817729166402825695630, 8.279681701868830168797054766565
