| L(s) = 1 | − 2·2-s + 3·3-s + 5·5-s − 6·6-s − 28·7-s + 8·8-s − 10·10-s + 15·11-s + 154·13-s + 56·14-s + 15·15-s − 16·16-s + 96·17-s + 37·19-s − 84·21-s − 30·22-s + 99·23-s + 24·24-s − 308·26-s − 27·27-s + 480·29-s − 30·30-s + 166·31-s + 45·33-s − 192·34-s − 140·35-s − 335·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.411·11-s + 3.28·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s + 1.36·17-s + 0.446·19-s − 0.872·21-s − 0.290·22-s + 0.897·23-s + 0.204·24-s − 2.32·26-s − 0.192·27-s + 3.07·29-s − 0.182·30-s + 0.961·31-s + 0.237·33-s − 0.968·34-s − 0.676·35-s − 1.48·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.653287155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.653287155\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 15 T - 1106 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 77 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 96 T + 4303 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 37 T - 5490 T^{2} - 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 99 T - 2366 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 240 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 166 T - 2235 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 335 T + 61572 T^{2} + 335 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 21 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 40 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 639 T + 304498 T^{2} - 639 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 153 T - 125468 T^{2} + 153 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 684 T + 262477 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 p T + 3 p^{2} T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 608 T + 68901 T^{2} + 608 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 198 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 338 T - 274773 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 736 T + 48657 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1290 T + 959131 T^{2} - 1290 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1456 T + p^{3} T^{2} )^{2} \) |
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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
−12.14568789761916355008581894741, −11.76930231146139983955585747815, −10.89099555937671581522216488306, −10.53928949101366059998445017688, −10.18083367419291481168623815169, −9.682888943569030021877590744476, −8.970631389394252210455440893873, −8.887016271059870094791153742060, −8.341604748086817554682780088377, −7.931403853639859081439534482436, −6.87529413658407331623882009737, −6.66170627101245203563872029088, −6.00289273615871157607039485175, −5.66705769591748871427503100805, −4.58254957367480121935402967494, −3.60143846654779505487521797602, −3.40697349544336486514821884317, −2.69049793668046771264096686170, −1.12490880894800414866483554584, −1.03454213067199664995345839378,
1.03454213067199664995345839378, 1.12490880894800414866483554584, 2.69049793668046771264096686170, 3.40697349544336486514821884317, 3.60143846654779505487521797602, 4.58254957367480121935402967494, 5.66705769591748871427503100805, 6.00289273615871157607039485175, 6.66170627101245203563872029088, 6.87529413658407331623882009737, 7.931403853639859081439534482436, 8.341604748086817554682780088377, 8.887016271059870094791153742060, 8.970631389394252210455440893873, 9.682888943569030021877590744476, 10.18083367419291481168623815169, 10.53928949101366059998445017688, 10.89099555937671581522216488306, 11.76930231146139983955585747815, 12.14568789761916355008581894741
