| L(s) = 1 | − 2·3-s + 2·7-s + 9-s + 6·13-s − 6·19-s − 4·21-s − 2·25-s + 4·27-s − 10·31-s − 8·37-s − 12·39-s − 2·49-s + 12·57-s + 12·61-s + 2·63-s − 6·67-s + 12·73-s + 4·75-s + 12·79-s − 11·81-s + 12·91-s + 20·93-s + 24·97-s + 16·103-s + 28·109-s + 16·111-s + 6·117-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s + 1.66·13-s − 1.37·19-s − 0.872·21-s − 2/5·25-s + 0.769·27-s − 1.79·31-s − 1.31·37-s − 1.92·39-s − 2/7·49-s + 1.58·57-s + 1.53·61-s + 0.251·63-s − 0.733·67-s + 1.40·73-s + 0.461·75-s + 1.35·79-s − 1.22·81-s + 1.25·91-s + 2.07·93-s + 2.43·97-s + 1.57·103-s + 2.68·109-s + 1.51·111-s + 0.554·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.114852345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114852345\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
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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
−8.619316920191259588988951873680, −8.348335745275121722939245092886, −7.74104708339414884017171728560, −7.20454848623926595633645086894, −6.69055955905639174645284330006, −6.17041088825483811045801141677, −5.97739832113511543571301006440, −5.31312871229344183136293557361, −4.99055990836743797546969625853, −4.38135387340709880605032405080, −3.70988945200155560355403264643, −3.39753357298482947412508202893, −2.14812974276536299924842384731, −1.70194718797425545200488150577, −0.63939406452837363732023448727,
0.63939406452837363732023448727, 1.70194718797425545200488150577, 2.14812974276536299924842384731, 3.39753357298482947412508202893, 3.70988945200155560355403264643, 4.38135387340709880605032405080, 4.99055990836743797546969625853, 5.31312871229344183136293557361, 5.97739832113511543571301006440, 6.17041088825483811045801141677, 6.69055955905639174645284330006, 7.20454848623926595633645086894, 7.74104708339414884017171728560, 8.348335745275121722939245092886, 8.619316920191259588988951873680
