| L(s) = 1 | − 4-s + 5-s − 5·9-s − 11-s + 16-s + 4·19-s − 20-s − 4·25-s − 5·29-s + 4·31-s + 5·36-s + 4·41-s + 44-s − 5·45-s − 55-s + 10·59-s + 4·61-s − 64-s − 71-s − 4·76-s + 20·79-s + 80-s + 16·81-s − 15·89-s + 4·95-s + 5·99-s + 4·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.447·5-s − 5/3·9-s − 0.301·11-s + 1/4·16-s + 0.917·19-s − 0.223·20-s − 4/5·25-s − 0.928·29-s + 0.718·31-s + 5/6·36-s + 0.624·41-s + 0.150·44-s − 0.745·45-s − 0.134·55-s + 1.30·59-s + 0.512·61-s − 1/8·64-s − 0.118·71-s − 0.458·76-s + 2.25·79-s + 0.111·80-s + 16/9·81-s − 1.58·89-s + 0.410·95-s + 0.502·99-s + 2/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5789095445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5789095445\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
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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
−13.51018298394949928685649002838, −12.90721272144625542771690814385, −12.07806623656308894598039569467, −11.53800051179541040464942422271, −11.00987828804187090531946099032, −10.15331403496232343322386834540, −9.513022230786289468619254879780, −8.959179379357012122284617774476, −8.209803403074668025415074567405, −7.63808546887048781562240619800, −6.47490536653561533931195767753, −5.62946018004322240278799245961, −5.20108476254412110565767497924, −3.80437078366842167705072317266, −2.63434450133726836876637171659,
2.63434450133726836876637171659, 3.80437078366842167705072317266, 5.20108476254412110565767497924, 5.62946018004322240278799245961, 6.47490536653561533931195767753, 7.63808546887048781562240619800, 8.209803403074668025415074567405, 8.959179379357012122284617774476, 9.513022230786289468619254879780, 10.15331403496232343322386834540, 11.00987828804187090531946099032, 11.53800051179541040464942422271, 12.07806623656308894598039569467, 12.90721272144625542771690814385, 13.51018298394949928685649002838
