L(s) = 1 | + 9·3-s − 6·5-s + 81·9-s − 108·11-s + 346·13-s − 54·15-s + 1.39e3·17-s + 1.01e3·19-s − 1.53e3·23-s − 3.08e3·25-s + 729·27-s − 3.76e3·29-s + 736·31-s − 972·33-s + 2.05e3·37-s + 3.11e3·39-s + 1.55e4·41-s + 1.10e4·43-s − 486·45-s − 4.56e3·47-s + 1.25e4·51-s − 7.96e3·53-s + 648·55-s + 9.10e3·57-s + 7.02e3·59-s − 2.68e4·61-s − 2.07e3·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.107·5-s + 1/3·9-s − 0.269·11-s + 0.567·13-s − 0.0619·15-s + 1.17·17-s + 0.643·19-s − 0.605·23-s − 0.988·25-s + 0.192·27-s − 0.830·29-s + 0.137·31-s − 0.155·33-s + 0.246·37-s + 0.327·39-s + 1.44·41-s + 0.910·43-s − 0.0357·45-s − 0.301·47-s + 0.677·51-s − 0.389·53-s + 0.0288·55-s + 0.371·57-s + 0.262·59-s − 0.924·61-s − 0.0609·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.815282936\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.815282936\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 6 T + p^{5} T^{2} \) |
| 11 | \( 1 + 108 T + p^{5} T^{2} \) |
| 13 | \( 1 - 346 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1398 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1012 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1536 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3762 T + p^{5} T^{2} \) |
| 31 | \( 1 - 736 T + p^{5} T^{2} \) |
| 37 | \( 1 - 2054 T + p^{5} T^{2} \) |
| 41 | \( 1 - 15534 T + p^{5} T^{2} \) |
| 43 | \( 1 - 11036 T + p^{5} T^{2} \) |
| 47 | \( 1 + 4560 T + p^{5} T^{2} \) |
| 53 | \( 1 + 7962 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7020 T + p^{5} T^{2} \) |
| 61 | \( 1 + 26870 T + p^{5} T^{2} \) |
| 67 | \( 1 - 52148 T + p^{5} T^{2} \) |
| 71 | \( 1 + 2544 T + p^{5} T^{2} \) |
| 73 | \( 1 - 9766 T + p^{5} T^{2} \) |
| 79 | \( 1 - 68672 T + p^{5} T^{2} \) |
| 83 | \( 1 - 61668 T + p^{5} T^{2} \) |
| 89 | \( 1 - 41454 T + p^{5} T^{2} \) |
| 97 | \( 1 - 111262 T + p^{5} T^{2} \) |
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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
−9.773083523589339363394772116270, −9.137834853713180035317126987273, −7.915915389194582265773586661602, −7.62738775552654160588568924937, −6.24438827215191373990848947083, −5.37379250372251904639380530811, −4.07002404696707268652485412090, −3.25409774896807186593900294124, −2.04397226402739159979435903329, −0.809272566730893268332528956863,
0.809272566730893268332528956863, 2.04397226402739159979435903329, 3.25409774896807186593900294124, 4.07002404696707268652485412090, 5.37379250372251904639380530811, 6.24438827215191373990848947083, 7.62738775552654160588568924937, 7.915915389194582265773586661602, 9.137834853713180035317126987273, 9.773083523589339363394772116270