| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s + 2·13-s + 14-s + 15-s + 16-s + 2·17-s − 18-s + 4·19-s − 20-s + 21-s − 4·22-s + 4·23-s + 24-s + 25-s − 2·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 249690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.413410540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.413410540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.65054732625733, −12.39496892887975, −11.66895858620162, −11.43560426331467, −11.13193669291218, −10.64888317983139, −9.787143681713511, −9.722363347472355, −9.291160459676276, −8.525902529967352, −8.344381787406108, −7.640730128253130, −7.077020844377407, −6.796132938125541, −6.382576114213188, −5.682466820584866, −5.313507027922444, −4.701527833636740, −3.950559432126578, −3.416575817682459, −3.253832596448907, −2.201430853156303, −1.541288740703335, −1.032689648972985, −0.4506876140972005,
0.4506876140972005, 1.032689648972985, 1.541288740703335, 2.201430853156303, 3.253832596448907, 3.416575817682459, 3.950559432126578, 4.701527833636740, 5.313507027922444, 5.682466820584866, 6.382576114213188, 6.796132938125541, 7.077020844377407, 7.640730128253130, 8.344381787406108, 8.525902529967352, 9.291160459676276, 9.722363347472355, 9.787143681713511, 10.64888317983139, 11.13193669291218, 11.43560426331467, 11.66895858620162, 12.39496892887975, 12.65054732625733
