L(s) = 1 | + 3·5-s − 2·13-s + 3·17-s + 4·19-s − 6·23-s + 4·25-s + 10·31-s − 7·37-s + 9·41-s + 5·43-s + 3·47-s − 6·53-s − 9·59-s − 8·61-s − 6·65-s + 8·67-s + 12·71-s + 10·73-s + 5·79-s + 9·83-s + 9·85-s + 18·89-s + 12·95-s − 8·97-s − 6·101-s + 10·103-s + 18·107-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 0.554·13-s + 0.727·17-s + 0.917·19-s − 1.25·23-s + 4/5·25-s + 1.79·31-s − 1.15·37-s + 1.40·41-s + 0.762·43-s + 0.437·47-s − 0.824·53-s − 1.17·59-s − 1.02·61-s − 0.744·65-s + 0.977·67-s + 1.42·71-s + 1.17·73-s + 0.562·79-s + 0.987·83-s + 0.976·85-s + 1.90·89-s + 1.23·95-s − 0.812·97-s − 0.597·101-s + 0.985·103-s + 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.687669561\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.687669561\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.021097364358920853575322970307, −7.62485920981535354564413333310, −6.54395789706067308742182699600, −6.06579030795155281178416264191, −5.35166810317699300434724489826, −4.70762208523864509811657203976, −3.62164001248046007628717958369, −2.67498057692181088842280910871, −1.95932865225983330371136953596, −0.911395083504015531197436683121,
0.911395083504015531197436683121, 1.95932865225983330371136953596, 2.67498057692181088842280910871, 3.62164001248046007628717958369, 4.70762208523864509811657203976, 5.35166810317699300434724489826, 6.06579030795155281178416264191, 6.54395789706067308742182699600, 7.62485920981535354564413333310, 8.021097364358920853575322970307