| L(s) = 1 | + 3·5-s − 9·13-s + 3·17-s − 8·19-s + 9·23-s + 5·25-s − 15·29-s + 8·31-s − 15·41-s + 21·43-s − 3·47-s + 2·49-s − 3·53-s − 3·59-s − 7·61-s − 27·65-s + 5·67-s + 9·71-s − 7·73-s + 7·79-s + 9·85-s − 15·89-s − 24·95-s + 15·97-s + 3·101-s − 32·103-s + 24·107-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 2.49·13-s + 0.727·17-s − 1.83·19-s + 1.87·23-s + 25-s − 2.78·29-s + 1.43·31-s − 2.34·41-s + 3.20·43-s − 0.437·47-s + 2/7·49-s − 0.412·53-s − 0.390·59-s − 0.896·61-s − 3.34·65-s + 0.610·67-s + 1.06·71-s − 0.819·73-s + 0.787·79-s + 0.976·85-s − 1.58·89-s − 2.46·95-s + 1.52·97-s + 0.298·101-s − 3.15·103-s + 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.853643001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.853643001\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T + 104 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−9.377632518712236069063980544897, −8.656697831805025522683660451173, −8.410683120107858467473488892818, −7.65203855491436737547510293610, −7.54791430450053274605739956041, −7.01561442683321049490596002548, −6.92043182240924466430954966606, −6.18451511083790265102550679048, −6.00882715940036594412783775695, −5.52660490685037210580707488045, −5.12321083206845880705186955617, −4.71025944107977331889811326204, −4.58022245359703675495449303907, −3.77485359380744971442191252173, −3.30812087854304377902864665616, −2.59360484349881165996791575665, −2.43972725055948611480604005300, −1.95757315298272901331529465462, −1.37392860530505182005739012755, −0.42366904640389536417051266681,
0.42366904640389536417051266681, 1.37392860530505182005739012755, 1.95757315298272901331529465462, 2.43972725055948611480604005300, 2.59360484349881165996791575665, 3.30812087854304377902864665616, 3.77485359380744971442191252173, 4.58022245359703675495449303907, 4.71025944107977331889811326204, 5.12321083206845880705186955617, 5.52660490685037210580707488045, 6.00882715940036594412783775695, 6.18451511083790265102550679048, 6.92043182240924466430954966606, 7.01561442683321049490596002548, 7.54791430450053274605739956041, 7.65203855491436737547510293610, 8.410683120107858467473488892818, 8.656697831805025522683660451173, 9.377632518712236069063980544897
