| L(s) = 1 | + 5-s − 13-s − 3·17-s − 4·23-s − 4·25-s + 29-s − 37-s + 9·41-s + 4·43-s + 4·47-s − 7·49-s − 3·53-s − 12·59-s + 14·61-s − 65-s + 16·67-s + 12·71-s + 10·73-s + 4·79-s − 3·85-s + 5·89-s − 5·97-s − 6·101-s + 8·103-s − 4·107-s + 11·109-s + 9·113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.277·13-s − 0.727·17-s − 0.834·23-s − 4/5·25-s + 0.185·29-s − 0.164·37-s + 1.40·41-s + 0.609·43-s + 0.583·47-s − 49-s − 0.412·53-s − 1.56·59-s + 1.79·61-s − 0.124·65-s + 1.95·67-s + 1.42·71-s + 1.17·73-s + 0.450·79-s − 0.325·85-s + 0.529·89-s − 0.507·97-s − 0.597·101-s + 0.788·103-s − 0.386·107-s + 1.05·109-s + 0.846·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.892570955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.892570955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.85986816927309772607959752044, −7.01845331256724419147793901890, −6.35446469945908884144434929599, −5.77174781419193029157109569710, −5.00827038610169503506569757074, −4.24593657871422489574674591940, −3.53665538818862108770554725859, −2.44242107430627669304745132058, −1.93557068353961077936810660016, −0.65264061506177647821101567632,
0.65264061506177647821101567632, 1.93557068353961077936810660016, 2.44242107430627669304745132058, 3.53665538818862108770554725859, 4.24593657871422489574674591940, 5.00827038610169503506569757074, 5.77174781419193029157109569710, 6.35446469945908884144434929599, 7.01845331256724419147793901890, 7.85986816927309772607959752044
