| L(s) = 1 | + 2.41·3-s − 5-s + 3.41·7-s + 2.82·9-s + 11-s + 2.24·13-s − 2.41·15-s + 3.41·17-s + 19-s + 8.24·21-s + 3·23-s − 4·25-s − 0.414·27-s − 6.24·29-s + 6.41·31-s + 2.41·33-s − 3.41·35-s + 10.0·37-s + 5.41·39-s − 1.65·41-s − 0.343·43-s − 2.82·45-s − 8.82·47-s + 4.65·49-s + 8.24·51-s − 4.48·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s − 0.447·5-s + 1.29·7-s + 0.942·9-s + 0.301·11-s + 0.621·13-s − 0.623·15-s + 0.828·17-s + 0.229·19-s + 1.79·21-s + 0.625·23-s − 0.800·25-s − 0.0797·27-s − 1.15·29-s + 1.15·31-s + 0.420·33-s − 0.577·35-s + 1.65·37-s + 0.866·39-s − 0.258·41-s − 0.0523·43-s − 0.421·45-s − 1.28·47-s + 0.665·49-s + 1.15·51-s − 0.616·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.654338004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.654338004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 + 0.343T + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.41T + 83T^{2} \) |
| 89 | \( 1 - 4.89T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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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
−8.356200451196195086732735778324, −7.983022249443780769423917231005, −7.61108929537963346105403834906, −6.51175753189090542551119841465, −5.48465051994605290669518655388, −4.57850545649442225980191650394, −3.80376312047561119603159084847, −3.11336695474722026611053655011, −2.06460366734261499283684678439, −1.18973822255237352751742223950,
1.18973822255237352751742223950, 2.06460366734261499283684678439, 3.11336695474722026611053655011, 3.80376312047561119603159084847, 4.57850545649442225980191650394, 5.48465051994605290669518655388, 6.51175753189090542551119841465, 7.61108929537963346105403834906, 7.983022249443780769423917231005, 8.356200451196195086732735778324
