| L(s) = 1 | + 4·5-s + 7-s − 6·9-s + 8·11-s + 4·13-s + 2·25-s − 16·31-s + 4·35-s + 8·43-s − 24·45-s + 16·47-s + 49-s + 32·55-s − 12·61-s − 6·63-s + 16·65-s + 8·67-s + 8·77-s + 27·81-s + 4·91-s − 48·99-s + 4·101-s + 32·103-s + 24·107-s + 4·113-s − 24·117-s + 26·121-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s − 2·9-s + 2.41·11-s + 1.10·13-s + 2/5·25-s − 2.87·31-s + 0.676·35-s + 1.21·43-s − 3.57·45-s + 2.33·47-s + 1/7·49-s + 4.31·55-s − 1.53·61-s − 0.755·63-s + 1.98·65-s + 0.977·67-s + 0.911·77-s + 3·81-s + 0.419·91-s − 4.82·99-s + 0.398·101-s + 3.15·103-s + 2.32·107-s + 0.376·113-s − 2.21·117-s + 2.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 351232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.818316330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.818316330\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.065821423802092395362181435296, −8.560607806232820418165457845354, −7.81893808816424475924755874673, −7.31356679414241138388846467495, −6.67949194679031572918581545842, −6.08808818165492598018676693090, −5.82613210142107824134972732669, −5.81038841174965326632002876390, −5.10093822894527537496894350174, −4.18465932715971087267449515138, −3.70152283742019411667382089257, −3.22234166202046004358507312699, −2.14556791802447766709825303506, −1.93975903006317803281774950018, −1.01536704926852438245703233735,
1.01536704926852438245703233735, 1.93975903006317803281774950018, 2.14556791802447766709825303506, 3.22234166202046004358507312699, 3.70152283742019411667382089257, 4.18465932715971087267449515138, 5.10093822894527537496894350174, 5.81038841174965326632002876390, 5.82613210142107824134972732669, 6.08808818165492598018676693090, 6.67949194679031572918581545842, 7.31356679414241138388846467495, 7.81893808816424475924755874673, 8.560607806232820418165457845354, 9.065821423802092395362181435296
