| L(s) = 1 | − 4-s + 2·11-s + 16-s + 8·19-s + 12·29-s + 16·31-s − 12·41-s − 2·44-s + 10·49-s + 16·61-s − 64-s − 12·71-s − 8·76-s − 28·79-s − 12·89-s − 12·101-s + 8·109-s − 12·116-s + 3·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.603·11-s + 1/4·16-s + 1.83·19-s + 2.22·29-s + 2.87·31-s − 1.87·41-s − 0.301·44-s + 10/7·49-s + 2.04·61-s − 1/8·64-s − 1.42·71-s − 0.917·76-s − 3.15·79-s − 1.27·89-s − 1.19·101-s + 0.766·109-s − 1.11·116-s + 3/11·121-s − 1.43·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.049702998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.049702998\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.664782909518632297295912105324, −8.238414568331591139577640142010, −7.78924244967713017272742860667, −7.16686295938038070670790008547, −7.10209218398243938230348323668, −6.66337877143013355261865359864, −6.32538742300598588344515614945, −5.80604571111637262822386563681, −5.58318888448732536257409824816, −5.02942439977974422593969569530, −4.79320397724313851884403347112, −4.26833418064369118492580917684, −4.16205507094380973754079920528, −3.44821593090815095482383272144, −2.98717809726799514704443415120, −2.85540869092853404967424156546, −2.25654632263624138682425558260, −1.25051033714618332800399908691, −1.23962517798520136727570334496, −0.55489720668283507282439958680,
0.55489720668283507282439958680, 1.23962517798520136727570334496, 1.25051033714618332800399908691, 2.25654632263624138682425558260, 2.85540869092853404967424156546, 2.98717809726799514704443415120, 3.44821593090815095482383272144, 4.16205507094380973754079920528, 4.26833418064369118492580917684, 4.79320397724313851884403347112, 5.02942439977974422593969569530, 5.58318888448732536257409824816, 5.80604571111637262822386563681, 6.32538742300598588344515614945, 6.66337877143013355261865359864, 7.10209218398243938230348323668, 7.16686295938038070670790008547, 7.78924244967713017272742860667, 8.238414568331591139577640142010, 8.664782909518632297295912105324
