| L(s) = 1 | + 3-s − 7-s − 2·9-s − 6·11-s + 5·13-s − 19-s − 21-s + 3·23-s − 5·25-s − 5·27-s − 9·29-s − 4·31-s − 6·33-s − 2·37-s + 5·39-s − 8·43-s − 6·49-s − 3·53-s − 57-s − 9·59-s + 10·61-s + 2·63-s − 5·67-s + 3·69-s − 6·71-s + 7·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.80·11-s + 1.38·13-s − 0.229·19-s − 0.218·21-s + 0.625·23-s − 25-s − 0.962·27-s − 1.67·29-s − 0.718·31-s − 1.04·33-s − 0.328·37-s + 0.800·39-s − 1.21·43-s − 6/7·49-s − 0.412·53-s − 0.132·57-s − 1.17·59-s + 1.28·61-s + 0.251·63-s − 0.610·67-s + 0.361·69-s − 0.712·71-s + 0.819·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2916800749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2916800749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.69689906603957, −13.41405686662163, −12.99909316590978, −12.73793980744086, −11.81220130213798, −11.31533831000028, −10.94538926117473, −10.48317618935484, −9.867894765091065, −9.319502179078990, −8.857311781669834, −8.309626178765610, −7.908262215570247, −7.495661450785631, −6.765745365478681, −6.114148312313132, −5.559390212451067, −5.311830533242366, −4.458789815333955, −3.581554877743056, −3.405007963317238, −2.717076622959414, −2.074919925400632, −1.457841846141822, −0.1561256662306693,
0.1561256662306693, 1.457841846141822, 2.074919925400632, 2.717076622959414, 3.405007963317238, 3.581554877743056, 4.458789815333955, 5.311830533242366, 5.559390212451067, 6.114148312313132, 6.765745365478681, 7.495661450785631, 7.908262215570247, 8.309626178765610, 8.857311781669834, 9.319502179078990, 9.867894765091065, 10.48317618935484, 10.94538926117473, 11.31533831000028, 11.81220130213798, 12.73793980744086, 12.99909316590978, 13.41405686662163, 13.69689906603957
