| L(s) = 1 | + 3.02·3-s + 2.82·5-s − 1.06·7-s + 6.12·9-s − 4.20·11-s + 3.16·13-s + 8.53·15-s + 5.45·17-s − 3.40·19-s − 3.20·21-s − 5.37·23-s + 2.97·25-s + 9.45·27-s + 4.95·29-s − 1.26·31-s − 12.7·33-s − 2.99·35-s + 0.990·37-s + 9.56·39-s − 3.45·41-s + 9.10·43-s + 17.3·45-s + 1.36·47-s − 5.87·49-s + 16.4·51-s + 3.14·53-s − 11.8·55-s + ⋯ |
| L(s) = 1 | + 1.74·3-s + 1.26·5-s − 0.401·7-s + 2.04·9-s − 1.26·11-s + 0.877·13-s + 2.20·15-s + 1.32·17-s − 0.782·19-s − 0.699·21-s − 1.12·23-s + 0.594·25-s + 1.81·27-s + 0.920·29-s − 0.227·31-s − 2.21·33-s − 0.506·35-s + 0.162·37-s + 1.53·39-s − 0.539·41-s + 1.38·43-s + 2.57·45-s + 0.199·47-s − 0.839·49-s + 2.30·51-s + 0.431·53-s − 1.60·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.600451406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.600451406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 - 3.02T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 1.06T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 + 3.40T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 - 4.95T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 0.990T + 37T^{2} \) |
| 41 | \( 1 + 3.45T + 41T^{2} \) |
| 43 | \( 1 - 9.10T + 43T^{2} \) |
| 47 | \( 1 - 1.36T + 47T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 + 3.67T + 61T^{2} \) |
| 67 | \( 1 + 8.87T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−9.654580110493719376522137851292, −8.768270701888513216516362555841, −8.152678165971751896467962555226, −7.45975850241372151924310212703, −6.28532528730064311369884109735, −5.56412667447968024918033826723, −4.25577139789676274874880898912, −3.19636612368354051107705773384, −2.50752871509487936923165976578, −1.57293867000652323425899183943,
1.57293867000652323425899183943, 2.50752871509487936923165976578, 3.19636612368354051107705773384, 4.25577139789676274874880898912, 5.56412667447968024918033826723, 6.28532528730064311369884109735, 7.45975850241372151924310212703, 8.152678165971751896467962555226, 8.768270701888513216516362555841, 9.654580110493719376522137851292
