L(s) = 1 | + 3·3-s + 2·7-s + 6·9-s − 11-s − 5·17-s − 19-s + 6·21-s + 2·23-s + 9·27-s − 8·29-s − 10·31-s − 3·33-s − 6·37-s + 3·41-s − 4·43-s + 4·47-s − 3·49-s − 15·51-s − 6·53-s − 3·57-s − 8·59-s + 10·61-s + 12·63-s − 67-s + 6·69-s + 12·71-s + 3·73-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 0.755·7-s + 2·9-s − 0.301·11-s − 1.21·17-s − 0.229·19-s + 1.30·21-s + 0.417·23-s + 1.73·27-s − 1.48·29-s − 1.79·31-s − 0.522·33-s − 0.986·37-s + 0.468·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 2.10·51-s − 0.824·53-s − 0.397·57-s − 1.04·59-s + 1.28·61-s + 1.51·63-s − 0.122·67-s + 0.722·69-s + 1.42·71-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.10679596501224, −14.70383804886225, −14.26557364578292, −13.78584099343883, −13.16046609322881, −12.90493016400786, −12.32663428651104, −11.34055391348467, −11.00864353772310, −10.49336178938037, −9.594012494578250, −9.284881700797515, −8.779139726744567, −8.262642759360846, −7.804263837341644, −7.182080429822773, −6.825766775796715, −5.807239024830680, −5.074928837517229, −4.507364294512272, −3.746530665734721, −3.405343613664947, −2.429878818804654, −2.025435328939351, −1.440847241613441, 0,
1.440847241613441, 2.025435328939351, 2.429878818804654, 3.405343613664947, 3.746530665734721, 4.507364294512272, 5.074928837517229, 5.807239024830680, 6.825766775796715, 7.182080429822773, 7.804263837341644, 8.262642759360846, 8.779139726744567, 9.284881700797515, 9.594012494578250, 10.49336178938037, 11.00864353772310, 11.34055391348467, 12.32663428651104, 12.90493016400786, 13.16046609322881, 13.78584099343883, 14.26557364578292, 14.70383804886225, 15.10679596501224