L(s) = 1 | − 4-s + 5·9-s − 6·11-s + 16-s + 10·19-s − 4·31-s − 5·36-s + 6·41-s + 6·44-s − 4·61-s − 64-s + 24·71-s − 10·76-s + 20·79-s + 16·81-s + 30·89-s − 30·99-s + 36·101-s + 20·109-s + 5·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + 149-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 5/3·9-s − 1.80·11-s + 1/4·16-s + 2.29·19-s − 0.718·31-s − 5/6·36-s + 0.937·41-s + 0.904·44-s − 0.512·61-s − 1/8·64-s + 2.84·71-s − 1.14·76-s + 2.25·79-s + 16/9·81-s + 3.17·89-s − 3.01·99-s + 3.58·101-s + 1.91·109-s + 5/11·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.615679209\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.615679209\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.210897878895161822577777184573, −8.945892250575963369894923740477, −8.111053233997072655314004902403, −7.931341697641231005226120585215, −7.51768774789389237729426652336, −7.49428889006010056802365393282, −6.97561977052450077694041106513, −6.39288274303282700210173489811, −6.06239305749795240628384771694, −5.30872003303464996801529057959, −5.24288503909713039923766936655, −4.86161418702765497160292727101, −4.48590962523561887033755428260, −3.79460407379841790905164282584, −3.44816921109119400284687205159, −3.11660283789478574423296050837, −2.21502812324535321828730172815, −2.04424886370320975969878846508, −1.03216917778632698643065556762, −0.67237267760842111817178513817,
0.67237267760842111817178513817, 1.03216917778632698643065556762, 2.04424886370320975969878846508, 2.21502812324535321828730172815, 3.11660283789478574423296050837, 3.44816921109119400284687205159, 3.79460407379841790905164282584, 4.48590962523561887033755428260, 4.86161418702765497160292727101, 5.24288503909713039923766936655, 5.30872003303464996801529057959, 6.06239305749795240628384771694, 6.39288274303282700210173489811, 6.97561977052450077694041106513, 7.49428889006010056802365393282, 7.51768774789389237729426652336, 7.931341697641231005226120585215, 8.111053233997072655314004902403, 8.945892250575963369894923740477, 9.210897878895161822577777184573