L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 11-s + 12-s + 6·13-s + 16-s + 17-s + 18-s + 8·19-s − 22-s + 8·23-s + 24-s + 6·26-s + 27-s + 6·29-s + 4·31-s + 32-s − 33-s + 34-s + 36-s − 10·37-s + 8·38-s + 6·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.66·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.83·19-s − 0.213·22-s + 1.66·23-s + 0.204·24-s + 1.17·26-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s − 1.64·37-s + 1.29·38-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.661348739\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.661348739\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.35608975175771, −14.49250517226358, −14.08678521673658, −13.70678556328310, −13.15581890923653, −12.78872501936430, −12.03717025885676, −11.48678185880629, −11.07423382940347, −10.34582142444737, −9.925085157820560, −9.107670222883537, −8.627689694425627, −8.105356241254875, −7.395126181386040, −6.870074348894907, −6.309704169167336, −5.469762899776719, −5.119458347454773, −4.331793316259028, −3.527741286247909, −3.174394093357881, −2.575288863613138, −1.403286039773526, −0.9988132644332563,
0.9988132644332563, 1.403286039773526, 2.575288863613138, 3.174394093357881, 3.527741286247909, 4.331793316259028, 5.119458347454773, 5.469762899776719, 6.309704169167336, 6.870074348894907, 7.395126181386040, 8.105356241254875, 8.627689694425627, 9.107670222883537, 9.925085157820560, 10.34582142444737, 11.07423382940347, 11.48678185880629, 12.03717025885676, 12.78872501936430, 13.15581890923653, 13.70678556328310, 14.08678521673658, 14.49250517226358, 15.35608975175771