L(s) = 1 | + 516·3-s + 1.05e4·5-s + 4.93e4·7-s + 8.91e4·9-s + 3.09e5·11-s + 1.72e6·13-s + 5.43e6·15-s − 2.27e6·17-s − 4.55e6·19-s + 2.54e7·21-s − 7.28e6·23-s + 6.20e7·25-s − 4.54e7·27-s + 6.90e7·29-s − 1.41e8·31-s + 1.59e8·33-s + 5.19e8·35-s − 7.11e8·37-s + 8.89e8·39-s − 1.22e9·41-s + 3.36e7·43-s + 9.38e8·45-s + 1.23e8·47-s + 4.53e8·49-s − 1.17e9·51-s − 1.10e9·53-s + 3.25e9·55-s + ⋯ |
L(s) = 1 | + 1.22·3-s + 1.50·5-s + 1.10·7-s + 0.503·9-s + 0.579·11-s + 1.28·13-s + 1.84·15-s − 0.389·17-s − 0.421·19-s + 1.35·21-s − 0.235·23-s + 1.27·25-s − 0.609·27-s + 0.625·29-s − 0.889·31-s + 0.710·33-s + 1.67·35-s − 1.68·37-s + 1.57·39-s − 1.65·41-s + 0.0348·43-s + 0.758·45-s + 0.0783·47-s + 0.229·49-s − 0.477·51-s − 0.363·53-s + 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(5.095357602\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.095357602\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 172 p T + p^{11} T^{2} \) |
| 5 | \( 1 - 2106 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 49304 T + p^{11} T^{2} \) |
| 11 | \( 1 - 309420 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1723594 T + p^{11} T^{2} \) |
| 17 | \( 1 + 2279502 T + p^{11} T^{2} \) |
| 19 | \( 1 + 4550444 T + p^{11} T^{2} \) |
| 23 | \( 1 + 7282872 T + p^{11} T^{2} \) |
| 29 | \( 1 - 69040026 T + p^{11} T^{2} \) |
| 31 | \( 1 + 141740704 T + p^{11} T^{2} \) |
| 37 | \( 1 + 711366974 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1225262214 T + p^{11} T^{2} \) |
| 43 | \( 1 - 781540 p T + p^{11} T^{2} \) |
| 47 | \( 1 - 123214608 T + p^{11} T^{2} \) |
| 53 | \( 1 + 1106121582 T + p^{11} T^{2} \) |
| 59 | \( 1 - 9062779932 T + p^{11} T^{2} \) |
| 61 | \( 1 - 3854150458 T + p^{11} T^{2} \) |
| 67 | \( 1 - 15313764676 T + p^{11} T^{2} \) |
| 71 | \( 1 - 20619626328 T + p^{11} T^{2} \) |
| 73 | \( 1 + 2063718694 T + p^{11} T^{2} \) |
| 79 | \( 1 - 13689871472 T + p^{11} T^{2} \) |
| 83 | \( 1 + 65570428908 T + p^{11} T^{2} \) |
| 89 | \( 1 + 29715508854 T + p^{11} T^{2} \) |
| 97 | \( 1 + 23439626206 T + p^{11} T^{2} \) |
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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
−13.04590531259448320816164025089, −11.33834022258209031358172459610, −10.12567158811118820610059134058, −8.906422307387044243899876667934, −8.366936548912076230098707416450, −6.64639225927700146410896491653, −5.32249591245726456227256774612, −3.68363562800224965500083306121, −2.15021801683291206221543622996, −1.47969466187915957762904013773,
1.47969466187915957762904013773, 2.15021801683291206221543622996, 3.68363562800224965500083306121, 5.32249591245726456227256774612, 6.64639225927700146410896491653, 8.366936548912076230098707416450, 8.906422307387044243899876667934, 10.12567158811118820610059134058, 11.33834022258209031358172459610, 13.04590531259448320816164025089