| L(s) = 1 | + 7-s − 4·9-s − 6·11-s + 2·23-s + 2·25-s + 6·29-s + 6·37-s − 14·43-s + 49-s + 4·53-s − 4·63-s + 4·67-s − 4·71-s − 6·77-s + 7·81-s + 24·99-s + 4·107-s + 30·109-s − 16·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 4/3·9-s − 1.80·11-s + 0.417·23-s + 2/5·25-s + 1.11·29-s + 0.986·37-s − 2.13·43-s + 1/7·49-s + 0.549·53-s − 0.503·63-s + 0.488·67-s − 0.474·71-s − 0.683·77-s + 7/9·81-s + 2.41·99-s + 0.386·107-s + 2.87·109-s − 1.50·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
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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
−7.895917488281601207766600652616, −7.34712762536545059633512441714, −6.87575675537425333213147181986, −6.40874182605597086759076354841, −5.88415764462973339147278271991, −5.48353800395172501182650654598, −5.08931505644412810836235000452, −4.75472740613494947367664163702, −4.21572364977097249304909065538, −3.26522287792153359976833622163, −3.08803541052695295072272863359, −2.50840604905452046589615206057, −2.01800254311245765313237641292, −0.937867358411644015673001852775, 0,
0.937867358411644015673001852775, 2.01800254311245765313237641292, 2.50840604905452046589615206057, 3.08803541052695295072272863359, 3.26522287792153359976833622163, 4.21572364977097249304909065538, 4.75472740613494947367664163702, 5.08931505644412810836235000452, 5.48353800395172501182650654598, 5.88415764462973339147278271991, 6.40874182605597086759076354841, 6.87575675537425333213147181986, 7.34712762536545059633512441714, 7.895917488281601207766600652616
