| L(s) = 1 | − 1.41·5-s + 4.24·7-s − 6·11-s + 5.65·13-s + 6·17-s + 4·19-s + 2.82·23-s − 2.99·25-s − 1.41·29-s + 1.41·31-s − 6·35-s − 8.48·37-s + 2·41-s + 2.82·47-s + 10.9·49-s + 9.89·53-s + 8.48·55-s − 4·59-s − 8.48·61-s − 8.00·65-s + 8·67-s − 2.82·71-s + 8·73-s − 25.4·77-s − 12.7·79-s − 2·83-s − 8.48·85-s + ⋯ |
| L(s) = 1 | − 0.632·5-s + 1.60·7-s − 1.80·11-s + 1.56·13-s + 1.45·17-s + 0.917·19-s + 0.589·23-s − 0.599·25-s − 0.262·29-s + 0.254·31-s − 1.01·35-s − 1.39·37-s + 0.312·41-s + 0.412·47-s + 1.57·49-s + 1.35·53-s + 1.14·55-s − 0.520·59-s − 1.08·61-s − 0.992·65-s + 0.977·67-s − 0.335·71-s + 0.936·73-s − 2.90·77-s − 1.43·79-s − 0.219·83-s − 0.920·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.173908043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.173908043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 9.89T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.276184087266556528535122354929, −7.63687164610730112986378270375, −7.28062220876485213823497085781, −5.79575944257530186179878689380, −5.41462593282243605796669919867, −4.70515435805137415906135440703, −3.73300385330888394602038847270, −3.00747817755691272887712103948, −1.80284531855813003661234071606, −0.866966124798289090124052354248,
0.866966124798289090124052354248, 1.80284531855813003661234071606, 3.00747817755691272887712103948, 3.73300385330888394602038847270, 4.70515435805137415906135440703, 5.41462593282243605796669919867, 5.79575944257530186179878689380, 7.28062220876485213823497085781, 7.63687164610730112986378270375, 8.276184087266556528535122354929
