| L(s) = 1 | − 5-s + 2·11-s − 5·17-s − 2·19-s + 2·23-s − 4·25-s + 10·29-s + 5·37-s − 3·41-s − 7·43-s + 3·47-s + 6·53-s − 2·55-s − 59-s + 6·61-s + 4·67-s + 8·71-s − 10·73-s − 3·79-s + 13·83-s + 5·85-s + 6·89-s + 2·95-s + 14·97-s − 10·101-s − 2·103-s − 18·107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s − 1.21·17-s − 0.458·19-s + 0.417·23-s − 4/5·25-s + 1.85·29-s + 0.821·37-s − 0.468·41-s − 1.06·43-s + 0.437·47-s + 0.824·53-s − 0.269·55-s − 0.130·59-s + 0.768·61-s + 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.337·79-s + 1.42·83-s + 0.542·85-s + 0.635·89-s + 0.205·95-s + 1.42·97-s − 0.995·101-s − 0.197·103-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
| \(L(1)\) |
\(\approx\) |
\(1.586678692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586678692\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.334513597865881309518697602844, −7.45880678156365974590710833510, −6.65724737036948039070918895032, −6.28089734425250307641247922927, −5.17261831698512850935825782097, −4.42843749256593256195863651803, −3.83480671395287911511255173794, −2.83109710285122323270485853335, −1.92056072863143155216859749459, −0.67526168700081318608927816074,
0.67526168700081318608927816074, 1.92056072863143155216859749459, 2.83109710285122323270485853335, 3.83480671395287911511255173794, 4.42843749256593256195863651803, 5.17261831698512850935825782097, 6.28089734425250307641247922927, 6.65724737036948039070918895032, 7.45880678156365974590710833510, 8.334513597865881309518697602844
