| L(s) = 1 | − 5-s + 13-s + 2·17-s − 8·23-s + 25-s − 29-s + 4·31-s + 2·37-s + 6·41-s + 12·43-s − 8·47-s − 7·49-s − 6·53-s + 12·59-s − 2·61-s − 65-s − 4·67-s − 2·73-s − 4·79-s − 4·83-s − 2·85-s − 10·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.277·13-s + 0.485·17-s − 1.66·23-s + 1/5·25-s − 0.185·29-s + 0.718·31-s + 0.328·37-s + 0.937·41-s + 1.82·43-s − 1.16·47-s − 49-s − 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.124·65-s − 0.488·67-s − 0.234·73-s − 0.450·79-s − 0.439·83-s − 0.216·85-s − 1.05·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265629776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265629776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.64099562021152, −12.35113237901111, −11.87159276660722, −11.39996153382146, −10.99908057913464, −10.53314633941267, −9.910153357351924, −9.630130327171567, −9.144226192705303, −8.409182572601759, −8.103320079961997, −7.758799845928059, −7.209823134687705, −6.569145985910270, −6.195986413762331, −5.625610567357989, −5.226543416621239, −4.377111574735182, −4.138967706436367, −3.652294074201162, −2.842398458684363, −2.563594379090643, −1.654108028252966, −1.183177264536669, −0.3160332935917858,
0.3160332935917858, 1.183177264536669, 1.654108028252966, 2.563594379090643, 2.842398458684363, 3.652294074201162, 4.138967706436367, 4.377111574735182, 5.226543416621239, 5.625610567357989, 6.195986413762331, 6.569145985910270, 7.209823134687705, 7.758799845928059, 8.103320079961997, 8.409182572601759, 9.144226192705303, 9.630130327171567, 9.910153357351924, 10.53314633941267, 10.99908057913464, 11.39996153382146, 11.87159276660722, 12.35113237901111, 12.64099562021152
