| L(s) = 1 | − 2·2-s + 5·3-s + 4·4-s − 10·6-s − 8·8-s − 2·9-s − 11-s + 20·12-s + 7·13-s + 16·16-s − 51·17-s + 4·18-s − 30·19-s + 2·22-s + 50·23-s − 40·24-s − 14·26-s − 145·27-s + 79·29-s + 212·31-s − 32·32-s − 5·33-s + 102·34-s − 8·36-s + 190·37-s + 60·38-s + 35·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.962·3-s + 1/2·4-s − 0.680·6-s − 0.353·8-s − 0.0740·9-s − 0.0274·11-s + 0.481·12-s + 0.149·13-s + 1/4·16-s − 0.727·17-s + 0.0523·18-s − 0.362·19-s + 0.0193·22-s + 0.453·23-s − 0.340·24-s − 0.105·26-s − 1.03·27-s + 0.505·29-s + 1.22·31-s − 0.176·32-s − 0.0263·33-s + 0.514·34-s − 0.0370·36-s + 0.844·37-s + 0.256·38-s + 0.143·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + T + p^{3} T^{2} \) |
| 13 | \( 1 - 7 T + p^{3} T^{2} \) |
| 17 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 30 T + p^{3} T^{2} \) |
| 23 | \( 1 - 50 T + p^{3} T^{2} \) |
| 29 | \( 1 - 79 T + p^{3} T^{2} \) |
| 31 | \( 1 - 212 T + p^{3} T^{2} \) |
| 37 | \( 1 - 190 T + p^{3} T^{2} \) |
| 41 | \( 1 - 308 T + p^{3} T^{2} \) |
| 43 | \( 1 + 422 T + p^{3} T^{2} \) |
| 47 | \( 1 - 121 T + p^{3} T^{2} \) |
| 53 | \( 1 + 664 T + p^{3} T^{2} \) |
| 59 | \( 1 + 628 T + p^{3} T^{2} \) |
| 61 | \( 1 - 684 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1056 T + p^{3} T^{2} \) |
| 71 | \( 1 - 744 T + p^{3} T^{2} \) |
| 73 | \( 1 - 726 T + p^{3} T^{2} \) |
| 79 | \( 1 + 407 T + p^{3} T^{2} \) |
| 83 | \( 1 - 644 T + p^{3} T^{2} \) |
| 89 | \( 1 - 880 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1351 T + p^{3} T^{2} \) |
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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
−8.135011771872096277764048218397, −7.88307018418132577573923475222, −6.73795795111803056445788388790, −6.18385178894876668102566405175, −4.99068394531414903316597545650, −4.00410660315680706014364912269, −2.96218149971273929077206352517, −2.38143665199504731280668083003, −1.26915815659263693740933515661, 0,
1.26915815659263693740933515661, 2.38143665199504731280668083003, 2.96218149971273929077206352517, 4.00410660315680706014364912269, 4.99068394531414903316597545650, 6.18385178894876668102566405175, 6.73795795111803056445788388790, 7.88307018418132577573923475222, 8.135011771872096277764048218397
