| L(s) = 1 | − 4·2-s + 29·3-s + 16·4-s − 116·6-s + 230·7-s − 64·8-s + 598·9-s + 121·11-s + 464·12-s − 112·13-s − 920·14-s + 256·16-s + 1.14e3·17-s − 2.39e3·18-s − 612·19-s + 6.67e3·21-s − 484·22-s + 1.94e3·23-s − 1.85e3·24-s + 448·26-s + 1.02e4·27-s + 3.68e3·28-s + 1.19e3·29-s − 1.03e3·31-s − 1.02e3·32-s + 3.50e3·33-s − 4.56e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.86·3-s + 1/2·4-s − 1.31·6-s + 1.77·7-s − 0.353·8-s + 2.46·9-s + 0.301·11-s + 0.930·12-s − 0.183·13-s − 1.25·14-s + 1/4·16-s + 0.958·17-s − 1.74·18-s − 0.388·19-s + 3.30·21-s − 0.213·22-s + 0.765·23-s − 0.657·24-s + 0.129·26-s + 2.71·27-s + 0.887·28-s + 0.263·29-s − 0.193·31-s − 0.176·32-s + 0.560·33-s − 0.677·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.573370496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.573370496\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 29 T + p^{5} T^{2} \) |
| 7 | \( 1 - 230 T + p^{5} T^{2} \) |
| 13 | \( 1 + 112 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1142 T + p^{5} T^{2} \) |
| 19 | \( 1 + 612 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1941 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1192 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1037 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8083 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10444 T + p^{5} T^{2} \) |
| 43 | \( 1 + 58 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8656 T + p^{5} T^{2} \) |
| 53 | \( 1 - 20318 T + p^{5} T^{2} \) |
| 59 | \( 1 + 21351 T + p^{5} T^{2} \) |
| 61 | \( 1 - 47044 T + p^{5} T^{2} \) |
| 67 | \( 1 + 48093 T + p^{5} T^{2} \) |
| 71 | \( 1 + 24967 T + p^{5} T^{2} \) |
| 73 | \( 1 - 42288 T + p^{5} T^{2} \) |
| 79 | \( 1 + 72410 T + p^{5} T^{2} \) |
| 83 | \( 1 - 15806 T + p^{5} T^{2} \) |
| 89 | \( 1 + 114761 T + p^{5} T^{2} \) |
| 97 | \( 1 - 5159 T + p^{5} T^{2} \) |
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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
−9.812159982116593362893011015017, −8.778602169194059161117056669872, −8.420573416234454948909027369281, −7.65921746914077168145223107814, −6.98474070466952244168611677980, −5.18601180409481052653832367908, −4.08802700269469330489906511024, −2.96235891418125549615666711904, −1.89083407067157941679845223933, −1.24138053693616527421602824271,
1.24138053693616527421602824271, 1.89083407067157941679845223933, 2.96235891418125549615666711904, 4.08802700269469330489906511024, 5.18601180409481052653832367908, 6.98474070466952244168611677980, 7.65921746914077168145223107814, 8.420573416234454948909027369281, 8.778602169194059161117056669872, 9.812159982116593362893011015017
