| L(s) = 1 | − 4-s + 2·5-s + 8·11-s + 16-s + 2·19-s − 2·20-s − 25-s − 4·29-s − 12·31-s − 8·44-s + 14·49-s + 16·55-s + 4·59-s − 12·61-s − 64-s + 24·71-s − 2·76-s + 28·79-s + 2·80-s + 8·89-s + 4·95-s + 100-s + 36·101-s − 8·109-s + 4·116-s + 26·121-s + 12·124-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s + 2.41·11-s + 1/4·16-s + 0.458·19-s − 0.447·20-s − 1/5·25-s − 0.742·29-s − 2.15·31-s − 1.20·44-s + 2·49-s + 2.15·55-s + 0.520·59-s − 1.53·61-s − 1/8·64-s + 2.84·71-s − 0.229·76-s + 3.15·79-s + 0.223·80-s + 0.847·89-s + 0.410·95-s + 1/10·100-s + 3.58·101-s − 0.766·109-s + 0.371·116-s + 2.36·121-s + 1.07·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.005552511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.005552511\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.478560357936067581335384702495, −9.228620766441018194027331559874, −8.890682772364043322594933188267, −8.619524224170295971456713097223, −7.79239889737717952551858614211, −7.65820989500424078068735344126, −7.03425856364710671627961810961, −6.75796566230088970954744596744, −6.18611663618006758365874689405, −6.01562802053806258529100842932, −5.47487895113361789651879457349, −5.10082158543761354992029439073, −4.59238737507768974909141457509, −3.92788901675125513669078175942, −3.56038146481319896683270141331, −3.53694432865273450749967203236, −2.30199979476279768089729394756, −2.00935667495296722648840420197, −1.36304288652075625360353109695, −0.71739516910332683208464858455,
0.71739516910332683208464858455, 1.36304288652075625360353109695, 2.00935667495296722648840420197, 2.30199979476279768089729394756, 3.53694432865273450749967203236, 3.56038146481319896683270141331, 3.92788901675125513669078175942, 4.59238737507768974909141457509, 5.10082158543761354992029439073, 5.47487895113361789651879457349, 6.01562802053806258529100842932, 6.18611663618006758365874689405, 6.75796566230088970954744596744, 7.03425856364710671627961810961, 7.65820989500424078068735344126, 7.79239889737717952551858614211, 8.619524224170295971456713097223, 8.890682772364043322594933188267, 9.228620766441018194027331559874, 9.478560357936067581335384702495
