| L(s) = 1 | − 2.59·3-s − 3.58·5-s − 4.34·7-s + 3.72·9-s − 3.81·11-s − 2.60·13-s + 9.29·15-s + 3.12·17-s − 1.39·19-s + 11.2·21-s + 7.84·25-s − 1.89·27-s + 4.34·29-s + 5.76·31-s + 9.88·33-s + 15.5·35-s − 7.02·37-s + 6.76·39-s − 1.45·41-s − 5.28·43-s − 13.3·45-s − 0.409·47-s + 11.8·49-s − 8.10·51-s + 10.6·53-s + 13.6·55-s + 3.61·57-s + ⋯ |
| L(s) = 1 | − 1.49·3-s − 1.60·5-s − 1.64·7-s + 1.24·9-s − 1.14·11-s − 0.723·13-s + 2.40·15-s + 0.758·17-s − 0.319·19-s + 2.45·21-s + 1.56·25-s − 0.363·27-s + 0.806·29-s + 1.03·31-s + 1.72·33-s + 2.62·35-s − 1.15·37-s + 1.08·39-s − 0.227·41-s − 0.806·43-s − 1.99·45-s − 0.0597·47-s + 1.69·49-s − 1.13·51-s + 1.46·53-s + 1.84·55-s + 0.478·57-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 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 7 | \( 1 + 4.34T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 29 | \( 1 - 4.34T + 29T^{2} \) |
| 31 | \( 1 - 5.76T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 + 0.409T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 2.14T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 6.75T + 67T^{2} \) |
| 71 | \( 1 + 0.889T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 6.69T + 89T^{2} \) |
| 97 | \( 1 + 2.05T + 97T^{2} \) |
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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
−7.927798015610145231911748874551, −6.96475830513982746452420563545, −6.79787515192519588861463721137, −5.75839720283370083921924188236, −5.11904807871469321753616993681, −4.34599299932412318227476888525, −3.45513345423528224307202865481, −2.72463219721051712822474252962, −0.67869560734292311864316731399, 0,
0.67869560734292311864316731399, 2.72463219721051712822474252962, 3.45513345423528224307202865481, 4.34599299932412318227476888525, 5.11904807871469321753616993681, 5.75839720283370083921924188236, 6.79787515192519588861463721137, 6.96475830513982746452420563545, 7.927798015610145231911748874551
