| L(s) = 1 | − 4-s + 2·9-s + 6·11-s + 16-s − 2·19-s − 12·29-s − 16·31-s − 2·36-s − 6·41-s − 6·44-s − 16·61-s − 64-s + 2·76-s + 20·79-s − 5·81-s + 12·89-s + 12·99-s − 24·101-s + 32·109-s + 12·116-s + 5·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s + 1.80·11-s + 1/4·16-s − 0.458·19-s − 2.22·29-s − 2.87·31-s − 1/3·36-s − 0.937·41-s − 0.904·44-s − 2.04·61-s − 1/8·64-s + 0.229·76-s + 2.25·79-s − 5/9·81-s + 1.27·89-s + 1.20·99-s − 2.38·101-s + 3.06·109-s + 1.11·116-s + 5/11·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.492004968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.492004968\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.434853445333141616286163736917, −8.921303883531473803900210860192, −8.550850246433368041702615164307, −7.79983141873359545484443776177, −7.66663578946429618992588832324, −7.19545151913731595438329629101, −6.92683881924668456693936582754, −6.30277971200210119303747218269, −6.21532830151476193457497773795, −5.45300789867343047372748065698, −5.34788663796958346446342040387, −4.71777690970606651764939440082, −4.20473625297410263714814984796, −3.88151128294011230237818422527, −3.58597229700582251951410040313, −3.20691277049366772666959683292, −2.19516494768558674776265842293, −1.66938604235428700514382012354, −1.50387593232407992736175011635, −0.41459514405894343044977775641,
0.41459514405894343044977775641, 1.50387593232407992736175011635, 1.66938604235428700514382012354, 2.19516494768558674776265842293, 3.20691277049366772666959683292, 3.58597229700582251951410040313, 3.88151128294011230237818422527, 4.20473625297410263714814984796, 4.71777690970606651764939440082, 5.34788663796958346446342040387, 5.45300789867343047372748065698, 6.21532830151476193457497773795, 6.30277971200210119303747218269, 6.92683881924668456693936582754, 7.19545151913731595438329629101, 7.66663578946429618992588832324, 7.79983141873359545484443776177, 8.550850246433368041702615164307, 8.921303883531473803900210860192, 9.434853445333141616286163736917
