| L(s) = 1 | + 2·2-s − 3-s + 7·5-s − 2·6-s − 8·8-s + 27·9-s + 14·10-s − 35·11-s − 132·13-s − 7·15-s − 16·16-s + 59·17-s + 54·18-s + 137·19-s − 70·22-s + 7·23-s + 8·24-s + 125·25-s − 264·26-s − 80·27-s + 212·29-s − 14·30-s + 75·31-s + 35·33-s + 118·34-s − 11·37-s + 274·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.192·3-s + 0.626·5-s − 0.136·6-s − 0.353·8-s + 9-s + 0.442·10-s − 0.959·11-s − 2.81·13-s − 0.120·15-s − 1/4·16-s + 0.841·17-s + 0.707·18-s + 1.65·19-s − 0.678·22-s + 0.0634·23-s + 0.0680·24-s + 25-s − 1.99·26-s − 0.570·27-s + 1.35·29-s − 0.0852·30-s + 0.434·31-s + 0.184·33-s + 0.595·34-s − 0.0488·37-s + 1.16·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.697515716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.697515716\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 26 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T - 76 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 35 T - 106 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 66 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 59 T - 1432 T^{2} - 59 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 137 T + 11910 T^{2} - 137 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 7 T - 12118 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 106 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 75 T - 24166 T^{2} - 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 11 T - 50532 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 498 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 260 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 171 T - 74582 T^{2} + 171 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 417 T + 25012 T^{2} - 417 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 17 T - 205090 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 51 T - 224380 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 439 T - 108042 T^{2} + 439 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 784 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 295 T - 301992 T^{2} - 295 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 495 T - 248014 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 932 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 873 T + 57160 T^{2} + 873 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 290 T + p^{3} T^{2} )^{2} \) |
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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
−14.08871258689931858751094523211, −12.85164898889506516891526159190, −12.83754641785020293197941347850, −12.23304108328036510648978664059, −11.92991772655909124463845465725, −11.04352786611972326818903308025, −10.32716730388733783864823702041, −9.825213500095759776591501489114, −9.701020819243641939159994261177, −8.958782792089562890978329087004, −7.80806810345649549707131232111, −7.33146557393786375184961152936, −7.16358221752932503464067860583, −5.81549740700953971713696179583, −5.57830663988827454210390884204, −4.62028414642847223046156168397, −4.51651740539359137357503085798, −2.89144620221992818889475911496, −2.54156757359145915234473106195, −0.896018045867936731791782438673,
0.896018045867936731791782438673, 2.54156757359145915234473106195, 2.89144620221992818889475911496, 4.51651740539359137357503085798, 4.62028414642847223046156168397, 5.57830663988827454210390884204, 5.81549740700953971713696179583, 7.16358221752932503464067860583, 7.33146557393786375184961152936, 7.80806810345649549707131232111, 8.958782792089562890978329087004, 9.701020819243641939159994261177, 9.825213500095759776591501489114, 10.32716730388733783864823702041, 11.04352786611972326818903308025, 11.92991772655909124463845465725, 12.23304108328036510648978664059, 12.83754641785020293197941347850, 12.85164898889506516891526159190, 14.08871258689931858751094523211
