| L(s) = 1 | − 2·5-s + 6·11-s + 12·13-s + 2·17-s + 4·19-s + 2·23-s + 5·25-s − 16·29-s + 4·31-s + 6·37-s + 20·41-s − 8·43-s + 4·47-s − 4·53-s − 12·55-s + 12·59-s − 2·61-s − 24·65-s − 12·67-s − 12·71-s − 2·73-s + 8·79-s − 4·85-s + 14·89-s − 8·95-s + 4·97-s + 18·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.80·11-s + 3.32·13-s + 0.485·17-s + 0.917·19-s + 0.417·23-s + 25-s − 2.97·29-s + 0.718·31-s + 0.986·37-s + 3.12·41-s − 1.21·43-s + 0.583·47-s − 0.549·53-s − 1.61·55-s + 1.56·59-s − 0.256·61-s − 2.97·65-s − 1.46·67-s − 1.42·71-s − 0.234·73-s + 0.900·79-s − 0.433·85-s + 1.48·89-s − 0.820·95-s + 0.406·97-s + 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.054592510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.054592510\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.990514810221194075889850625207, −8.295023896267385061275607708472, −8.029577197452804059379952354159, −7.74793813406754934169506697800, −7.27970938946927070859648463986, −6.89615936411685841124371654939, −6.57122699526027125603996781185, −5.90349585021183599700280251037, −5.86638544713334092593953757412, −5.74366097791981921822208983570, −4.79613993214622136243609025512, −4.39548419357018262452162177714, −3.94107821959265308267700049238, −3.75438122028500785591262019069, −3.39764029631657180243475095437, −3.08850448664679287822549028900, −2.20535994662372310825988280562, −1.42925876137627701846308689913, −1.16829104384579676334603675886, −0.75090183608207605383098160578,
0.75090183608207605383098160578, 1.16829104384579676334603675886, 1.42925876137627701846308689913, 2.20535994662372310825988280562, 3.08850448664679287822549028900, 3.39764029631657180243475095437, 3.75438122028500785591262019069, 3.94107821959265308267700049238, 4.39548419357018262452162177714, 4.79613993214622136243609025512, 5.74366097791981921822208983570, 5.86638544713334092593953757412, 5.90349585021183599700280251037, 6.57122699526027125603996781185, 6.89615936411685841124371654939, 7.27970938946927070859648463986, 7.74793813406754934169506697800, 8.029577197452804059379952354159, 8.295023896267385061275607708472, 8.990514810221194075889850625207
