| L(s) = 1 | + 9·3-s + 80·7-s + 81·9-s + 684·11-s + 978·13-s + 862·17-s + 916·19-s + 720·21-s + 1.55e3·23-s + 729·27-s − 7.31e3·29-s − 9.31e3·31-s + 6.15e3·33-s + 8.82e3·37-s + 8.80e3·39-s − 3.28e3·41-s − 7.55e3·43-s + 5.96e3·47-s − 1.04e4·49-s + 7.75e3·51-s + 8.69e3·53-s + 8.24e3·57-s − 4.20e4·59-s + 3.75e4·61-s + 6.48e3·63-s − 2.93e4·67-s + 1.39e4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.617·7-s + 1/3·9-s + 1.70·11-s + 1.60·13-s + 0.723·17-s + 0.582·19-s + 0.356·21-s + 0.611·23-s + 0.192·27-s − 1.61·29-s − 1.74·31-s + 0.984·33-s + 1.05·37-s + 0.926·39-s − 0.305·41-s − 0.623·43-s + 0.393·47-s − 0.619·49-s + 0.417·51-s + 0.425·53-s + 0.336·57-s − 1.57·59-s + 1.29·61-s + 0.205·63-s − 0.798·67-s + 0.353·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.018233145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.018233145\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 80 T + p^{5} T^{2} \) |
| 11 | \( 1 - 684 T + p^{5} T^{2} \) |
| 13 | \( 1 - 978 T + p^{5} T^{2} \) |
| 17 | \( 1 - 862 T + p^{5} T^{2} \) |
| 19 | \( 1 - 916 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1552 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7314 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9312 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8826 T + p^{5} T^{2} \) |
| 41 | \( 1 + 3286 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7556 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5960 T + p^{5} T^{2} \) |
| 53 | \( 1 - 8698 T + p^{5} T^{2} \) |
| 59 | \( 1 + 42036 T + p^{5} T^{2} \) |
| 61 | \( 1 - 37518 T + p^{5} T^{2} \) |
| 67 | \( 1 + 29324 T + p^{5} T^{2} \) |
| 71 | \( 1 - 84408 T + p^{5} T^{2} \) |
| 73 | \( 1 - 46550 T + p^{5} T^{2} \) |
| 79 | \( 1 - 26752 T + p^{5} T^{2} \) |
| 83 | \( 1 - 7956 T + p^{5} T^{2} \) |
| 89 | \( 1 - 59674 T + p^{5} T^{2} \) |
| 97 | \( 1 + 136898 T + p^{5} T^{2} \) |
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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
−9.561837485842592351015719079478, −9.083783281079767277602859281337, −8.209718034351718817922422889662, −7.31411826938067185819851643395, −6.32692849030153219449589783202, −5.32694473408731700339080377970, −3.93610865555768492360179681899, −3.45598110410793175982515838798, −1.75984338237498964797833473259, −1.07593525641134169258923820977,
1.07593525641134169258923820977, 1.75984338237498964797833473259, 3.45598110410793175982515838798, 3.93610865555768492360179681899, 5.32694473408731700339080377970, 6.32692849030153219449589783202, 7.31411826938067185819851643395, 8.209718034351718817922422889662, 9.083783281079767277602859281337, 9.561837485842592351015719079478
