| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s + 8·6-s + 8·9-s − 12·11-s − 8·12-s − 4·16-s − 8·17-s − 16·18-s + 24·22-s − 12·27-s + 8·32-s + 48·33-s + 16·34-s + 16·36-s − 12·41-s − 12·43-s − 24·44-s + 16·48-s + 32·51-s + 24·54-s − 8·64-s − 96·66-s + 12·67-s − 16·68-s − 24·73-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s + 3.26·6-s + 8/3·9-s − 3.61·11-s − 2.30·12-s − 16-s − 1.94·17-s − 3.77·18-s + 5.11·22-s − 2.30·27-s + 1.41·32-s + 8.35·33-s + 2.74·34-s + 8/3·36-s − 1.87·41-s − 1.82·43-s − 3.61·44-s + 2.30·48-s + 4.48·51-s + 3.26·54-s − 64-s − 11.8·66-s + 1.46·67-s − 1.94·68-s − 2.80·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.86543843523725153069004658553, −11.51002869178452917126363312877, −10.99296282352174022324289171520, −10.79435693539942004941676460471, −10.25028305894285021328916407748, −10.19208672449491571158597493896, −9.468078530657130202143325614055, −8.536829961717031401451695817128, −8.239392002889202088141311778017, −7.71703435992756767366116558877, −6.90045596591976007798566833346, −6.80108311468957659426418931131, −5.86748235504678641306859102020, −5.31096702673219580020982660690, −5.01538615927079748835865910659, −4.44058643613850798526754055740, −2.82311213713830116792009835201, −1.94465861321050862254360420318, 0, 0,
1.94465861321050862254360420318, 2.82311213713830116792009835201, 4.44058643613850798526754055740, 5.01538615927079748835865910659, 5.31096702673219580020982660690, 5.86748235504678641306859102020, 6.80108311468957659426418931131, 6.90045596591976007798566833346, 7.71703435992756767366116558877, 8.239392002889202088141311778017, 8.536829961717031401451695817128, 9.468078530657130202143325614055, 10.19208672449491571158597493896, 10.25028305894285021328916407748, 10.79435693539942004941676460471, 10.99296282352174022324289171520, 11.51002869178452917126363312877, 11.86543843523725153069004658553
