| L(s) = 1 | + 9-s + 6·11-s + 2·19-s − 8·29-s − 4·31-s + 10·41-s − 6·49-s + 8·59-s − 4·61-s + 16·71-s + 4·79-s − 8·81-s + 10·89-s + 6·99-s + 4·101-s − 20·109-s + 9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 1.80·11-s + 0.458·19-s − 1.48·29-s − 0.718·31-s + 1.56·41-s − 6/7·49-s + 1.04·59-s − 0.512·61-s + 1.89·71-s + 0.450·79-s − 8/9·81-s + 1.05·89-s + 0.603·99-s + 0.398·101-s − 1.91·109-s + 9/11·121-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 + 0.0783·163-s + 0.0773·167-s − 1.69·169-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
| \(L(1)\) |
\(\approx\) |
\(1.470959946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470959946\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
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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
−10.20180815861356361490370091265, −9.582033606717249363947775982558, −9.278219465883864885661113329083, −8.963854504971669405945035421263, −8.125582488418344144044469805908, −7.64200687670396379296472795399, −7.00969980928139886854855183907, −6.60985208785318643839630817275, −5.93402856145694718744873210504, −5.38624637278194682152779051948, −4.54725818598329256564023811581, −3.87971842666786877152833262703, −3.46380958781975248637069890164, −2.22950249091185203532348235563, −1.26135690971352753974238485933,
1.26135690971352753974238485933, 2.22950249091185203532348235563, 3.46380958781975248637069890164, 3.87971842666786877152833262703, 4.54725818598329256564023811581, 5.38624637278194682152779051948, 5.93402856145694718744873210504, 6.60985208785318643839630817275, 7.00969980928139886854855183907, 7.64200687670396379296472795399, 8.125582488418344144044469805908, 8.963854504971669405945035421263, 9.278219465883864885661113329083, 9.582033606717249363947775982558, 10.20180815861356361490370091265
