| L(s) = 1 | − 4·13-s − 6·25-s − 4·37-s − 10·49-s + 28·61-s + 20·73-s − 20·97-s + 12·109-s − 6·121-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 + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.10·13-s − 6/5·25-s − 0.657·37-s − 1.42·49-s + 3.58·61-s + 2.34·73-s − 2.03·97-s + 1.14·109-s − 0.545·121-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 + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.369899334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369899334\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.988692551879225765942246529287, −7.54414099658816705393057461976, −7.11637967987278977626078878884, −6.66142997878964053618256415130, −6.36897077911175274709355626908, −5.67589470598115755067299597557, −5.24252067499370010106342493266, −5.01404649196971011738958367359, −4.33704338614765596619172810543, −3.80602529089120251316754004693, −3.45457617540241674630539161201, −2.61856629445227575193132115840, −2.24251629150705986170173266597, −1.57160529682628953082929600681, −0.50410049751026002221381444165,
0.50410049751026002221381444165, 1.57160529682628953082929600681, 2.24251629150705986170173266597, 2.61856629445227575193132115840, 3.45457617540241674630539161201, 3.80602529089120251316754004693, 4.33704338614765596619172810543, 5.01404649196971011738958367359, 5.24252067499370010106342493266, 5.67589470598115755067299597557, 6.36897077911175274709355626908, 6.66142997878964053618256415130, 7.11637967987278977626078878884, 7.54414099658816705393057461976, 7.988692551879225765942246529287
