| L(s) = 1 | + 2·3-s − 4-s + 2·7-s + 3·9-s + 6·11-s − 2·12-s + 16-s + 4·21-s − 25-s + 4·27-s − 2·28-s + 12·33-s − 3·36-s + 2·37-s + 18·41-s − 6·44-s + 2·48-s − 11·49-s + 6·53-s + 6·63-s − 64-s − 28·67-s + 12·71-s − 4·73-s − 2·75-s + 12·77-s + 5·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 0.755·7-s + 9-s + 1.80·11-s − 0.577·12-s + 1/4·16-s + 0.872·21-s − 1/5·25-s + 0.769·27-s − 0.377·28-s + 2.08·33-s − 1/2·36-s + 0.328·37-s + 2.81·41-s − 0.904·44-s + 0.288·48-s − 1.57·49-s + 0.824·53-s + 0.755·63-s − 1/8·64-s − 3.42·67-s + 1.42·71-s − 0.468·73-s − 0.230·75-s + 1.36·77-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.039200573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.039200573\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 185 T^{2} + p^{2} T^{4} \) |
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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
−9.778889492193505775411039001439, −9.631179766256328939191660575874, −8.990531378553575466977013585560, −8.948766992817877193727099784717, −8.588620419311570080910708482829, −7.986929831024541833911621527324, −7.56693447289936871167308672863, −7.45025400592873956382054631167, −6.75450091698385634105245396914, −6.19958673141326957642421202709, −6.00267136238734659368426683644, −5.22311361208433505588912027052, −4.66570496297784145510553134022, −4.18735030223877815230402845466, −4.06626180075827773134897064500, −3.35216953744300208575858342703, −2.87689287993356520111263435646, −2.11479306499460418659692072182, −1.54219227692156980844826263577, −0.930205145806927511101236992244,
0.930205145806927511101236992244, 1.54219227692156980844826263577, 2.11479306499460418659692072182, 2.87689287993356520111263435646, 3.35216953744300208575858342703, 4.06626180075827773134897064500, 4.18735030223877815230402845466, 4.66570496297784145510553134022, 5.22311361208433505588912027052, 6.00267136238734659368426683644, 6.19958673141326957642421202709, 6.75450091698385634105245396914, 7.45025400592873956382054631167, 7.56693447289936871167308672863, 7.986929831024541833911621527324, 8.588620419311570080910708482829, 8.948766992817877193727099784717, 8.990531378553575466977013585560, 9.631179766256328939191660575874, 9.778889492193505775411039001439
