| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 4·8-s + 2·11-s + 2·12-s + 7·13-s + 8·16-s − 4·17-s − 5·19-s + 4·22-s − 4·23-s + 4·24-s + 14·26-s − 27-s + 2·29-s + 2·31-s + 8·32-s + 2·33-s − 8·34-s − 6·37-s − 10·38-s + 7·39-s + 8·43-s + 4·44-s − 8·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1.41·8-s + 0.603·11-s + 0.577·12-s + 1.94·13-s + 2·16-s − 0.970·17-s − 1.14·19-s + 0.852·22-s − 0.834·23-s + 0.816·24-s + 2.74·26-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 1.41·32-s + 0.348·33-s − 1.37·34-s − 0.986·37-s − 1.62·38-s + 1.12·39-s + 1.21·43-s + 0.603·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.922142176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.922142176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−10.49883312814910141838710549122, −9.795395106254966682717986417707, −9.302664835513665809146268575937, −8.966181447470414873305967393314, −8.382416756725084400928295558550, −8.119119395370405718523480914819, −7.88554528712028068521958289174, −7.02850842515923628510680732015, −6.69464177711858945238913960720, −6.16358440798796608697724520946, −6.14212350635257311956021965237, −5.22115972504531713903081601284, −4.99254610332728267157180307001, −4.20458072790890249736225063431, −4.02802319064685982637872588994, −3.66806343905495960337243802677, −3.14450093339916925812852119327, −2.20636428958873047271220317465, −1.90107900311943424066202625823, −1.01142801531687217733300863554,
1.01142801531687217733300863554, 1.90107900311943424066202625823, 2.20636428958873047271220317465, 3.14450093339916925812852119327, 3.66806343905495960337243802677, 4.02802319064685982637872588994, 4.20458072790890249736225063431, 4.99254610332728267157180307001, 5.22115972504531713903081601284, 6.14212350635257311956021965237, 6.16358440798796608697724520946, 6.69464177711858945238913960720, 7.02850842515923628510680732015, 7.88554528712028068521958289174, 8.119119395370405718523480914819, 8.382416756725084400928295558550, 8.966181447470414873305967393314, 9.302664835513665809146268575937, 9.795395106254966682717986417707, 10.49883312814910141838710549122
