| L(s) = 1 | − 3·9-s − 4·13-s + 8·17-s − 10·29-s − 12·37-s − 10·41-s − 7·49-s + 4·53-s − 10·61-s + 16·73-s + 9·81-s − 10·89-s − 8·97-s + 2·101-s − 6·109-s − 16·113-s + 12·117-s + ⋯ |
| L(s) = 1 | − 9-s − 1.10·13-s + 1.94·17-s − 1.85·29-s − 1.97·37-s − 1.56·41-s − 49-s + 0.549·53-s − 1.28·61-s + 1.87·73-s + 81-s − 1.05·89-s − 0.812·97-s + 0.199·101-s − 0.574·109-s − 1.50·113-s + 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.066578188161773859679921125938, −8.126003367030777913782292064447, −7.52991763396308675946168049663, −6.62707438006867438574420001208, −5.43538712614250917522273729751, −5.22283258119030602960852633346, −3.70896411876379581085476870424, −2.98396765796017704987428125492, −1.73719838151596358495089080076, 0,
1.73719838151596358495089080076, 2.98396765796017704987428125492, 3.70896411876379581085476870424, 5.22283258119030602960852633346, 5.43538712614250917522273729751, 6.62707438006867438574420001208, 7.52991763396308675946168049663, 8.126003367030777913782292064447, 9.066578188161773859679921125938
