| L(s) = 1 | + 24·7-s − 9·9-s + 124·17-s − 144·23-s + 186·25-s − 408·31-s − 44·41-s − 1.20e3·47-s − 254·49-s − 216·63-s + 912·71-s + 1.64e3·73-s − 2.71e3·79-s + 81·81-s − 1.87e3·89-s + 2.55e3·97-s + 1.89e3·103-s − 1.24e3·113-s + 2.97e3·119-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.11e3·153-s + ⋯ |
| L(s) = 1 | + 1.29·7-s − 1/3·9-s + 1.76·17-s − 1.30·23-s + 1.48·25-s − 2.36·31-s − 0.167·41-s − 3.72·47-s − 0.740·49-s − 0.431·63-s + 1.52·71-s + 2.63·73-s − 3.86·79-s + 1/9·81-s − 2.23·89-s + 2.67·97-s + 1.81·103-s − 1.03·113-s + 2.29·119-s + 1.89·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.589·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.570074885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.570074885\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 186 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2518 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2054 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 32394 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 204 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49322 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 117398 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 600 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 232218 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 274826 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 446906 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 480422 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 456 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 822 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1356 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1131910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 938 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1278 T + p^{3} T^{2} )^{2} \) |
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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
−11.25320203167392078872742371362, −10.78179207922824643398379039771, −10.22665350304568739322843965755, −9.723481108191528504881255951692, −9.489656622657277754922864419143, −8.597579411705327839531431692221, −8.404868040358122708562215616763, −7.86072404831049335806811520909, −7.63449617440326613744674874774, −6.87394291216297848290382686869, −6.45355276658753861853258837915, −5.58354550709204744473542521018, −5.40228291509979591401147309975, −4.83957795552222977033539825418, −4.26240616441031676735322052055, −3.31499842296347331767532362132, −3.19128049737484532564874046418, −1.78702475744060968774524541995, −1.71040998256064115654498497226, −0.54872275098143309599137984574,
0.54872275098143309599137984574, 1.71040998256064115654498497226, 1.78702475744060968774524541995, 3.19128049737484532564874046418, 3.31499842296347331767532362132, 4.26240616441031676735322052055, 4.83957795552222977033539825418, 5.40228291509979591401147309975, 5.58354550709204744473542521018, 6.45355276658753861853258837915, 6.87394291216297848290382686869, 7.63449617440326613744674874774, 7.86072404831049335806811520909, 8.404868040358122708562215616763, 8.597579411705327839531431692221, 9.489656622657277754922864419143, 9.723481108191528504881255951692, 10.22665350304568739322843965755, 10.78179207922824643398379039771, 11.25320203167392078872742371362
