| L(s) = 1 | + 4-s + 7-s + 5·9-s + 16-s − 10·19-s − 23-s − 4·25-s + 28-s − 3·29-s + 5·36-s − 6·37-s − 2·41-s − 43-s − 2·47-s − 4·49-s − 14·53-s − 21·59-s + 5·61-s + 5·63-s + 64-s − 10·76-s + 16·81-s + 6·83-s − 92-s − 97-s − 4·100-s + 11·101-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.377·7-s + 5/3·9-s + 1/4·16-s − 2.29·19-s − 0.208·23-s − 4/5·25-s + 0.188·28-s − 0.557·29-s + 5/6·36-s − 0.986·37-s − 0.312·41-s − 0.152·43-s − 0.291·47-s − 4/7·49-s − 1.92·53-s − 2.73·59-s + 0.640·61-s + 0.629·63-s + 1/8·64-s − 1.14·76-s + 16/9·81-s + 0.658·83-s − 0.104·92-s − 0.101·97-s − 2/5·100-s + 1.09·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1086652 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1086652 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 197 | $C_2$ | \( 1 + 15 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.937489870297617962370796073197, −7.40688863714518191553238730783, −6.99416262411676456265123010966, −6.52225762526998144182689484854, −6.22215146706749026424409341903, −5.80717622196040195289809596383, −4.96623393555267967121930872194, −4.60404139128366083187817467248, −4.33269874194742680896479410821, −3.61328671076668705180369337377, −3.27820317400924173927982383440, −2.18708248345597179437495781862, −1.91079761193276615339679270933, −1.40455365999027045766594357835, 0,
1.40455365999027045766594357835, 1.91079761193276615339679270933, 2.18708248345597179437495781862, 3.27820317400924173927982383440, 3.61328671076668705180369337377, 4.33269874194742680896479410821, 4.60404139128366083187817467248, 4.96623393555267967121930872194, 5.80717622196040195289809596383, 6.22215146706749026424409341903, 6.52225762526998144182689484854, 6.99416262411676456265123010966, 7.40688863714518191553238730783, 7.937489870297617962370796073197
