| L(s) = 1 | − 1.41·5-s + 4·7-s − 5.65·11-s + 4.24·13-s + 6·17-s − 5.65·19-s − 8·23-s − 2.99·25-s + 4.24·29-s + 4·31-s − 5.65·35-s + 1.41·37-s + 2·41-s + 5.65·43-s + 8·47-s + 9·49-s − 9.89·53-s + 8.00·55-s + 4.24·61-s − 6·65-s − 11.3·67-s + 10·73-s − 22.6·77-s + 12·79-s − 5.65·83-s − 8.48·85-s − 16·89-s + ⋯ |
| L(s) = 1 | − 0.632·5-s + 1.51·7-s − 1.70·11-s + 1.17·13-s + 1.45·17-s − 1.29·19-s − 1.66·23-s − 0.599·25-s + 0.787·29-s + 0.718·31-s − 0.956·35-s + 0.232·37-s + 0.312·41-s + 0.862·43-s + 1.16·47-s + 1.28·49-s − 1.35·53-s + 1.07·55-s + 0.543·61-s − 0.744·65-s − 1.38·67-s + 1.17·73-s − 2.57·77-s + 1.35·79-s − 0.620·83-s − 0.920·85-s − 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.922897704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922897704\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 9.89T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−7.910682317044871260829505317169, −7.38514722401851373782687364737, −6.06830685205524415599789091866, −5.77989820400681881910117058570, −4.82984541207375552991913941282, −4.31215356241840517518563529909, −3.55495426200283822248518196174, −2.52996513911009628824153563103, −1.76571358898074508641665096787, −0.67281993067179701427344904710,
0.67281993067179701427344904710, 1.76571358898074508641665096787, 2.52996513911009628824153563103, 3.55495426200283822248518196174, 4.31215356241840517518563529909, 4.82984541207375552991913941282, 5.77989820400681881910117058570, 6.06830685205524415599789091866, 7.38514722401851373782687364737, 7.910682317044871260829505317169
