| L(s) = 1 | − 4·5-s + 5·9-s − 2·11-s − 8·19-s + 11·25-s + 2·29-s + 12·31-s + 20·41-s − 20·45-s + 8·55-s + 12·59-s + 8·61-s − 32·71-s + 22·79-s + 16·81-s + 24·89-s + 32·95-s − 10·99-s + 30·109-s − 19·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 5/3·9-s − 0.603·11-s − 1.83·19-s + 11/5·25-s + 0.371·29-s + 2.15·31-s + 3.12·41-s − 2.98·45-s + 1.07·55-s + 1.56·59-s + 1.02·61-s − 3.79·71-s + 2.47·79-s + 16/9·81-s + 2.54·89-s + 3.28·95-s − 1.00·99-s + 2.87·109-s − 1.72·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.779246143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.779246143\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
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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
−9.222422968800396224412836216707, −8.942764804799642424343641100245, −8.536677345833429951867614839856, −8.004807196360362903996573534881, −7.87531002249553070606636250104, −7.52800915619795128245450998189, −7.06836063240099952703718723183, −6.78346923116214863583088022875, −6.13082086016116177721583784792, −6.12099001509499741255267768934, −5.06130092412691465277068948515, −4.78967635232577585948693001257, −4.30783774599762127389943430788, −4.15514975812277907812288964179, −3.79022341719444720541010835395, −3.06396650645317043376460911432, −2.54131173833178097868470506294, −2.05328371863018114736947674458, −1.02379750656731340639668614838, −0.60425216495940544797012983440,
0.60425216495940544797012983440, 1.02379750656731340639668614838, 2.05328371863018114736947674458, 2.54131173833178097868470506294, 3.06396650645317043376460911432, 3.79022341719444720541010835395, 4.15514975812277907812288964179, 4.30783774599762127389943430788, 4.78967635232577585948693001257, 5.06130092412691465277068948515, 6.12099001509499741255267768934, 6.13082086016116177721583784792, 6.78346923116214863583088022875, 7.06836063240099952703718723183, 7.52800915619795128245450998189, 7.87531002249553070606636250104, 8.004807196360362903996573534881, 8.536677345833429951867614839856, 8.942764804799642424343641100245, 9.222422968800396224412836216707
