| L(s) = 1 | − 4·5-s + 8·11-s + 14·19-s + 11·25-s + 16·29-s + 12·31-s + 2·41-s − 11·49-s − 32·55-s + 6·59-s − 20·61-s + 8·71-s − 8·79-s + 36·89-s − 56·95-s − 12·101-s + 24·109-s + 26·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s − 48·155-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2.41·11-s + 3.21·19-s + 11/5·25-s + 2.97·29-s + 2.15·31-s + 0.312·41-s − 1.57·49-s − 4.31·55-s + 0.781·59-s − 2.56·61-s + 0.949·71-s − 0.900·79-s + 3.81·89-s − 5.74·95-s − 1.19·101-s + 2.29·109-s + 2.36·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s − 3.85·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.146343996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.146343996\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.942849650722421386068305288317, −8.276762096577964556977062174990, −8.122363914215625404509134907329, −7.77579063357639884124368843227, −7.38033383980527697006230033056, −7.02511229787380944644404450095, −6.54902217043252638088100068813, −6.37100990275788189639824573582, −6.07702857582457450779528451176, −5.09748967704765382424665637500, −4.99346641996509709966668455258, −4.48572990879663811298662254339, −4.26461682711633204992295064241, −3.65594157796337916906888451346, −3.23988083663932460683747761151, −3.14350044318165741166100186621, −2.53587172005214100289893803273, −1.32487325532347433708638408481, −1.11993476585825243543626752281, −0.72551168355155893685546516186,
0.72551168355155893685546516186, 1.11993476585825243543626752281, 1.32487325532347433708638408481, 2.53587172005214100289893803273, 3.14350044318165741166100186621, 3.23988083663932460683747761151, 3.65594157796337916906888451346, 4.26461682711633204992295064241, 4.48572990879663811298662254339, 4.99346641996509709966668455258, 5.09748967704765382424665637500, 6.07702857582457450779528451176, 6.37100990275788189639824573582, 6.54902217043252638088100068813, 7.02511229787380944644404450095, 7.38033383980527697006230033056, 7.77579063357639884124368843227, 8.122363914215625404509134907329, 8.276762096577964556977062174990, 8.942849650722421386068305288317
