| L(s) = 1 | − 1.56·3-s + 2.56·7-s − 0.561·9-s + 11-s + 0.561·13-s − 5.56·17-s − 3·19-s − 4·21-s + 23-s + 5.56·27-s + 1.43·29-s + 5.12·31-s − 1.56·33-s − 3.12·37-s − 0.876·39-s − 1.87·41-s + 7.68·43-s + 6·47-s − 0.438·49-s + 8.68·51-s − 9.12·53-s + 4.68·57-s − 4·59-s + 2.24·61-s − 1.43·63-s − 5.56·67-s − 1.56·69-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.968·7-s − 0.187·9-s + 0.301·11-s + 0.155·13-s − 1.34·17-s − 0.688·19-s − 0.872·21-s + 0.208·23-s + 1.07·27-s + 0.267·29-s + 0.920·31-s − 0.271·33-s − 0.513·37-s − 0.140·39-s − 0.293·41-s + 1.17·43-s + 0.875·47-s − 0.0626·49-s + 1.21·51-s − 1.25·53-s + 0.620·57-s − 0.520·59-s + 0.287·61-s − 0.181·63-s − 0.679·67-s − 0.187·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 9.12T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 + 5.56T + 67T^{2} \) |
| 71 | \( 1 + 1.12T + 71T^{2} \) |
| 73 | \( 1 - 6.12T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 - 6.12T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−7.27092848330043864981750836766, −6.52687848430293233086844458956, −6.09322956478084460671637428084, −5.27977461887273242641463275075, −4.62017918192067456788682050577, −4.17606689321520368742281394589, −2.95535430058418892422802130602, −2.09085422950409717680777800459, −1.13576512018787292269590161908, 0,
1.13576512018787292269590161908, 2.09085422950409717680777800459, 2.95535430058418892422802130602, 4.17606689321520368742281394589, 4.62017918192067456788682050577, 5.27977461887273242641463275075, 6.09322956478084460671637428084, 6.52687848430293233086844458956, 7.27092848330043864981750836766
