| L(s) = 1 | + 4-s − 2·9-s − 6·11-s + 16-s + 4·19-s − 5·25-s + 6·29-s − 3·31-s − 2·36-s − 6·44-s + 14·49-s − 12·59-s − 2·61-s + 64-s − 12·71-s + 4·76-s + 16·79-s − 5·81-s + 12·99-s − 5·100-s + 24·101-s + 4·109-s + 6·116-s + 14·121-s − 3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2/3·9-s − 1.80·11-s + 1/4·16-s + 0.917·19-s − 25-s + 1.11·29-s − 0.538·31-s − 1/3·36-s − 0.904·44-s + 2·49-s − 1.56·59-s − 0.256·61-s + 1/8·64-s − 1.42·71-s + 0.458·76-s + 1.80·79-s − 5/9·81-s + 1.20·99-s − 1/2·100-s + 2.38·101-s + 0.383·109-s + 0.557·116-s + 1.27·121-s − 0.269·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7443062799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7443062799\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−12.81365678367467052347521989621, −11.98862070987432293690846708722, −11.78573272186784587161989779031, −10.80280655722587761529649694641, −10.57296840298900291185541341501, −9.861315886434960457037564566044, −9.082208952973575690302975869099, −8.294667149989560448194403193013, −7.69342434658164905742698545991, −7.20803473701915876196808011938, −6.05879722313541813624414510500, −5.56326081307184092647987072175, −4.71663520601975594839256349202, −3.31085421966376311394103008228, −2.42821548860184841917021394637,
2.42821548860184841917021394637, 3.31085421966376311394103008228, 4.71663520601975594839256349202, 5.56326081307184092647987072175, 6.05879722313541813624414510500, 7.20803473701915876196808011938, 7.69342434658164905742698545991, 8.294667149989560448194403193013, 9.082208952973575690302975869099, 9.861315886434960457037564566044, 10.57296840298900291185541341501, 10.80280655722587761529649694641, 11.78573272186784587161989779031, 11.98862070987432293690846708722, 12.81365678367467052347521989621
