L(s) = 1 | + 1.25e3·7-s − 2.00e3·13-s + 8.61e4·19-s + 7.81e4·25-s − 1.78e5·31-s + 6.71e5·37-s − 1.03e6·43-s + 8.23e5·49-s − 1.99e6·61-s − 4.44e6·67-s + 1.00e7·73-s + 4.51e6·79-s − 2.52e6·91-s + 1.75e7·97-s − 1.38e7·103-s − 3.36e7·109-s + 1.94e7·121-s + 127-s + 131-s + 1.08e8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 1.38·7-s − 0.253·13-s + 2.88·19-s + 25-s − 1.07·31-s + 2.17·37-s − 1.98·43-s + 49-s − 1.12·61-s − 1.80·67-s + 3.03·73-s + 1.03·79-s − 0.350·91-s + 1.94·97-s − 1.25·103-s − 2.48·109-s + 121-s + 3.98·133-s + 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(4.615057429\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.615057429\) |
\(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 1763 T + p^{7} T^{2} )( 1 + 508 T + p^{7} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 12605 T + p^{7} T^{2} )( 1 + 14614 T + p^{7} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 43091 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 152471 T + p^{7} T^{2} )( 1 + 331387 T + p^{7} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 335663 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 409495 T + p^{7} T^{2} )( 1 + 625729 T + p^{7} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1537199 T + p^{7} T^{2} )( 1 + 3535546 T + p^{7} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 385072 T + p^{7} T^{2} )( 1 + 4058455 T + p^{7} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5038001 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8763044 T + p^{7} T^{2} )( 1 + 4245427 T + p^{7} T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12245198 T + p^{7} T^{2} )( 1 - 5276357 T + p^{7} T^{2} ) \) |
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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
−10.78154424339096840007699820101, −10.19578184185024158343331482357, −9.548968360856021141180483906759, −9.333162125645891107221925913040, −8.858479070282314912540175090805, −7.982831491062107186195680955757, −7.896343572717068397663009302978, −7.49090394515011331040934004210, −6.87419629459664287245596170862, −6.32044502932621520721026689136, −5.55762644764613117732483870055, −5.04782341872770680526287062310, −4.98814032405941765433406111486, −4.17595760059411888740140581175, −3.46543167398601003265889903963, −2.96652544479717029990442757968, −2.30917772549947749673124595328, −1.45925848908918509599536895654, −1.19437947030151818747507394182, −0.50125250960161671365432263149,
0.50125250960161671365432263149, 1.19437947030151818747507394182, 1.45925848908918509599536895654, 2.30917772549947749673124595328, 2.96652544479717029990442757968, 3.46543167398601003265889903963, 4.17595760059411888740140581175, 4.98814032405941765433406111486, 5.04782341872770680526287062310, 5.55762644764613117732483870055, 6.32044502932621520721026689136, 6.87419629459664287245596170862, 7.49090394515011331040934004210, 7.896343572717068397663009302978, 7.982831491062107186195680955757, 8.858479070282314912540175090805, 9.333162125645891107221925913040, 9.548968360856021141180483906759, 10.19578184185024158343331482357, 10.78154424339096840007699820101