| L(s) = 1 | − 4-s − 9-s + 4·11-s + 16-s − 16·19-s + 8·29-s − 4·31-s + 36-s − 2·41-s − 4·44-s − 2·49-s − 30·59-s + 10·61-s − 64-s − 2·71-s + 16·76-s − 24·79-s + 81-s + 4·89-s − 4·99-s − 20·101-s + 4·109-s − 8·116-s − 10·121-s + 4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1.20·11-s + 1/4·16-s − 3.67·19-s + 1.48·29-s − 0.718·31-s + 1/6·36-s − 0.312·41-s − 0.603·44-s − 2/7·49-s − 3.90·59-s + 1.28·61-s − 1/8·64-s − 0.237·71-s + 1.83·76-s − 2.70·79-s + 1/9·81-s + 0.423·89-s − 0.402·99-s − 1.99·101-s + 0.383·109-s − 0.742·116-s − 0.909·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6517793818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6517793818\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−8.914147617683957278269128815251, −8.699014297837687460777474835280, −8.513266821007712667320951346169, −8.039339131730361769055049848285, −7.72210094517840288875288565481, −7.02716953861322278269096910719, −6.65404525500654111465341399049, −6.47707174733260893400457293623, −6.04916120122282390631626381735, −5.74148591225785876529250480653, −5.08735119759958399972964687477, −4.55627179694115357332180529950, −4.20433673111469906716071038403, −4.20069463002536196815285492343, −3.42797119317678671543979844038, −2.95063153621646548487442655277, −2.38480716153883640442766421381, −1.78058518827819546049342350544, −1.34996548618509835653470766168, −0.27030444657839510129600609241,
0.27030444657839510129600609241, 1.34996548618509835653470766168, 1.78058518827819546049342350544, 2.38480716153883640442766421381, 2.95063153621646548487442655277, 3.42797119317678671543979844038, 4.20069463002536196815285492343, 4.20433673111469906716071038403, 4.55627179694115357332180529950, 5.08735119759958399972964687477, 5.74148591225785876529250480653, 6.04916120122282390631626381735, 6.47707174733260893400457293623, 6.65404525500654111465341399049, 7.02716953861322278269096910719, 7.72210094517840288875288565481, 8.039339131730361769055049848285, 8.513266821007712667320951346169, 8.699014297837687460777474835280, 8.914147617683957278269128815251
