| L(s) = 1 | + 12·17-s + 8·19-s − 10·25-s − 20·31-s + 2·49-s + 24·59-s + 20·61-s − 8·67-s + 24·71-s − 4·73-s + 20·79-s − 4·103-s + 24·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.91·17-s + 1.83·19-s − 2·25-s − 3.59·31-s + 2/7·49-s + 3.12·59-s + 2.56·61-s − 0.977·67-s + 2.84·71-s − 0.468·73-s + 2.25·79-s − 0.394·103-s + 2.32·107-s + 0.909·121-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 + 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.034953180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.034953180\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.073208049576535529011561465060, −8.590067112621452960464968356750, −8.112699725932733472101803620504, −7.900893763137617359634014820226, −7.40872140924841305327437322265, −7.25925218713316289330286749062, −6.99540040482353581622120047140, −6.19899197619047223189784051360, −5.71869382657016986079824858758, −5.54401857037134018691675973596, −5.26239202779875142383189519877, −5.01675021272084989323685143065, −3.93938276919960824219396381649, −3.69312256337309369945735400237, −3.56272381534382604322226150596, −3.09558569138143154285089528654, −2.09727542061268540924599918882, −2.02687537929811228345164510534, −1.09623745212119098968117753023, −0.65922392756904491271660784896,
0.65922392756904491271660784896, 1.09623745212119098968117753023, 2.02687537929811228345164510534, 2.09727542061268540924599918882, 3.09558569138143154285089528654, 3.56272381534382604322226150596, 3.69312256337309369945735400237, 3.93938276919960824219396381649, 5.01675021272084989323685143065, 5.26239202779875142383189519877, 5.54401857037134018691675973596, 5.71869382657016986079824858758, 6.19899197619047223189784051360, 6.99540040482353581622120047140, 7.25925218713316289330286749062, 7.40872140924841305327437322265, 7.900893763137617359634014820226, 8.112699725932733472101803620504, 8.590067112621452960464968356750, 9.073208049576535529011561465060
