| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 2·13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 20-s + 21-s − 22-s − 3·23-s + 24-s + 25-s − 2·26-s − 27-s − 28-s + 3·29-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 + 0.301·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s − 0.213·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5149304607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5149304607\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.71573259032368, −12.09699394881057, −11.71664717351916, −11.34276457570608, −10.96078843105657, −10.34723624887778, −9.950132073529063, −9.686519096993824, −8.969933560954861, −8.490809636582879, −8.232527296201035, −7.578118774906627, −7.071056184955169, −6.672352895836361, −6.279119760267525, −5.641526699235606, −5.277702999106341, −4.558218622829358, −3.960674041214104, −3.538283897093203, −2.970779236350339, −2.236386493627046, −1.552820269004814, −1.049146948619373, −0.2533994066621246,
0.2533994066621246, 1.049146948619373, 1.552820269004814, 2.236386493627046, 2.970779236350339, 3.538283897093203, 3.960674041214104, 4.558218622829358, 5.277702999106341, 5.641526699235606, 6.279119760267525, 6.672352895836361, 7.071056184955169, 7.578118774906627, 8.232527296201035, 8.490809636582879, 8.969933560954861, 9.686519096993824, 9.950132073529063, 10.34723624887778, 10.96078843105657, 11.34276457570608, 11.71664717351916, 12.09699394881057, 12.71573259032368
