| L(s) = 1 | + 4·2-s − 24·3-s + 16·4-s − 96·6-s + 172·7-s + 64·8-s + 333·9-s + 132·11-s − 384·12-s + 946·13-s + 688·14-s + 256·16-s + 222·17-s + 1.33e3·18-s + 500·19-s − 4.12e3·21-s + 528·22-s − 3.56e3·23-s − 1.53e3·24-s + 3.78e3·26-s − 2.16e3·27-s + 2.75e3·28-s + 2.19e3·29-s + 2.31e3·31-s + 1.02e3·32-s − 3.16e3·33-s + 888·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.53·3-s + 1/2·4-s − 1.08·6-s + 1.32·7-s + 0.353·8-s + 1.37·9-s + 0.328·11-s − 0.769·12-s + 1.55·13-s + 0.938·14-s + 1/4·16-s + 0.186·17-s + 0.968·18-s + 0.317·19-s − 2.04·21-s + 0.232·22-s − 1.40·23-s − 0.544·24-s + 1.09·26-s − 0.570·27-s + 0.663·28-s + 0.483·29-s + 0.432·31-s + 0.176·32-s − 0.506·33-s + 0.131·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.862557969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.862557969\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 8 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 172 T + p^{5} T^{2} \) |
| 11 | \( 1 - 12 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 946 T + p^{5} T^{2} \) |
| 17 | \( 1 - 222 T + p^{5} T^{2} \) |
| 19 | \( 1 - 500 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3564 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2190 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2312 T + p^{5} T^{2} \) |
| 37 | \( 1 - 11242 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1242 T + p^{5} T^{2} \) |
| 43 | \( 1 + 20624 T + p^{5} T^{2} \) |
| 47 | \( 1 + 6588 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21066 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7980 T + p^{5} T^{2} \) |
| 61 | \( 1 - 16622 T + p^{5} T^{2} \) |
| 67 | \( 1 + 1808 T + p^{5} T^{2} \) |
| 71 | \( 1 + 24528 T + p^{5} T^{2} \) |
| 73 | \( 1 + 20474 T + p^{5} T^{2} \) |
| 79 | \( 1 + 46240 T + p^{5} T^{2} \) |
| 83 | \( 1 - 51576 T + p^{5} T^{2} \) |
| 89 | \( 1 + 110310 T + p^{5} T^{2} \) |
| 97 | \( 1 - 78382 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.48483808848266451335504399132, −13.32477201402923938865987275390, −11.84295292830749377577200305128, −11.44388509266639529732734034938, −10.37442758285016075606394534399, −8.150373610245563647085011739279, −6.45709258718860244390328613144, −5.46274576827229743989482644355, −4.22940074327482037024575411839, −1.30148043880589658682887465765,
1.30148043880589658682887465765, 4.22940074327482037024575411839, 5.46274576827229743989482644355, 6.45709258718860244390328613144, 8.150373610245563647085011739279, 10.37442758285016075606394534399, 11.44388509266639529732734034938, 11.84295292830749377577200305128, 13.32477201402923938865987275390, 14.48483808848266451335504399132
