| L(s) = 1 | + 1.37·5-s − 3.37·7-s + 1.37·11-s + 2·13-s − 1.37·17-s − 19-s − 8.74·23-s − 3.11·25-s − 2.74·29-s + 6.74·31-s − 4.62·35-s + 4.74·37-s − 3.37·43-s − 13.3·47-s + 4.37·49-s + 2.74·53-s + 1.88·55-s − 2.62·61-s + 2.74·65-s + 9.48·67-s − 12·71-s − 5.37·73-s − 4.62·77-s − 8·79-s + 8.74·83-s − 1.88·85-s − 14.7·89-s + ⋯ |
| L(s) = 1 | + 0.613·5-s − 1.27·7-s + 0.413·11-s + 0.554·13-s − 0.332·17-s − 0.229·19-s − 1.82·23-s − 0.623·25-s − 0.509·29-s + 1.21·31-s − 0.782·35-s + 0.780·37-s − 0.514·43-s − 1.95·47-s + 0.624·49-s + 0.376·53-s + 0.253·55-s − 0.336·61-s + 0.340·65-s + 1.15·67-s − 1.42·71-s − 0.628·73-s − 0.527·77-s − 0.900·79-s + 0.959·83-s − 0.204·85-s − 1.56·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 3.37T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 2.74T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 5.37T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 8.74T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−8.491889982084046144249139008598, −7.75637007124788722904540893489, −6.54923593566407051824565505092, −6.32565818442085182321759233482, −5.57873378733363067590571790105, −4.33755312834980223530662977223, −3.61874723484322138287259115311, −2.63598484346465082372558884813, −1.59639676538250773195616108165, 0,
1.59639676538250773195616108165, 2.63598484346465082372558884813, 3.61874723484322138287259115311, 4.33755312834980223530662977223, 5.57873378733363067590571790105, 6.32565818442085182321759233482, 6.54923593566407051824565505092, 7.75637007124788722904540893489, 8.491889982084046144249139008598
