| L(s) = 1 | − 4·5-s + 6·13-s − 8·17-s + 5·25-s + 4·29-s + 2·37-s − 16·41-s − 14·49-s − 8·53-s + 10·61-s − 24·65-s − 6·73-s + 32·85-s − 16·89-s − 36·97-s − 40·101-s + 6·109-s − 16·113-s − 22·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.66·13-s − 1.94·17-s + 25-s + 0.742·29-s + 0.328·37-s − 2.49·41-s − 2·49-s − 1.09·53-s + 1.28·61-s − 2.97·65-s − 0.702·73-s + 3.47·85-s − 1.69·89-s − 3.65·97-s − 3.98·101-s + 0.574·109-s − 1.50·113-s − 2·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−8.549664320534920130452322868605, −8.379859570239159023629443917060, −7.940983534529901900071170250520, −7.82336450262115715202956535392, −6.98266470829251783263143876777, −6.79763238598130107172210796561, −6.55323386207423441565482066239, −6.19772515397853326248217132886, −5.40040882629185586874703660705, −5.21342169130791343157784205613, −4.47241613502914019846643175068, −4.32102035360810672281463146472, −3.74822689864830892315566360089, −3.69624076068292913409317372452, −2.90937050589233368890153506241, −2.63528933529652172863687198528, −1.58953134256885053149055926902, −1.35113073394202297598788621155, 0, 0,
1.35113073394202297598788621155, 1.58953134256885053149055926902, 2.63528933529652172863687198528, 2.90937050589233368890153506241, 3.69624076068292913409317372452, 3.74822689864830892315566360089, 4.32102035360810672281463146472, 4.47241613502914019846643175068, 5.21342169130791343157784205613, 5.40040882629185586874703660705, 6.19772515397853326248217132886, 6.55323386207423441565482066239, 6.79763238598130107172210796561, 6.98266470829251783263143876777, 7.82336450262115715202956535392, 7.940983534529901900071170250520, 8.379859570239159023629443917060, 8.549664320534920130452322868605
