| L(s) = 1 | + 1.99·3-s + 1.23·5-s − 1.08·7-s + 0.971·9-s − 5.84·11-s + 2.45·13-s + 2.46·15-s − 4.32·17-s − 0.00965·19-s − 2.15·21-s − 3.46·25-s − 4.04·27-s − 0.0882·29-s + 2.58·31-s − 11.6·33-s − 1.34·35-s − 0.117·37-s + 4.89·39-s − 5.81·41-s − 2.55·43-s + 1.20·45-s + 8.99·47-s − 5.82·49-s − 8.62·51-s − 14.1·53-s − 7.23·55-s − 0.0192·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.553·5-s − 0.409·7-s + 0.323·9-s − 1.76·11-s + 0.681·13-s + 0.636·15-s − 1.04·17-s − 0.00221·19-s − 0.471·21-s − 0.693·25-s − 0.778·27-s − 0.0163·29-s + 0.464·31-s − 2.02·33-s − 0.226·35-s − 0.0193·37-s + 0.784·39-s − 0.908·41-s − 0.389·43-s + 0.179·45-s + 1.31·47-s − 0.832·49-s − 1.20·51-s − 1.94·53-s − 0.975·55-s − 0.00254·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.99T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 + 5.84T + 11T^{2} \) |
| 13 | \( 1 - 2.45T + 13T^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 + 0.00965T + 19T^{2} \) |
| 29 | \( 1 + 0.0882T + 29T^{2} \) |
| 31 | \( 1 - 2.58T + 31T^{2} \) |
| 37 | \( 1 + 0.117T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 + 2.55T + 43T^{2} \) |
| 47 | \( 1 - 8.99T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 1.80T + 59T^{2} \) |
| 61 | \( 1 - 8.76T + 61T^{2} \) |
| 67 | \( 1 + 4.80T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 9.93T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 0.334T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.222597833936783133648300560079, −7.49637084827356010190848258339, −6.59607470805897370082255740979, −5.82694580060012641215645281936, −5.07926177021719301508360868806, −4.07522223849886653823207471895, −3.14053168050270338051731146729, −2.55419763473426918614172119150, −1.78338800412453402089141130063, 0,
1.78338800412453402089141130063, 2.55419763473426918614172119150, 3.14053168050270338051731146729, 4.07522223849886653823207471895, 5.07926177021719301508360868806, 5.82694580060012641215645281936, 6.59607470805897370082255740979, 7.49637084827356010190848258339, 8.222597833936783133648300560079
