L(s) = 1 | − 3·5-s − 3·9-s − 11-s + 6·17-s + 19-s − 3·23-s + 4·25-s + 6·29-s − 2·31-s − 6·41-s + 9·43-s + 9·45-s + 3·47-s − 2·53-s + 3·55-s + 12·59-s − 11·61-s − 8·67-s + 6·71-s + 15·73-s − 12·79-s + 9·81-s − 7·83-s − 18·85-s + 8·89-s − 3·95-s + 6·97-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 9-s − 0.301·11-s + 1.45·17-s + 0.229·19-s − 0.625·23-s + 4/5·25-s + 1.11·29-s − 0.359·31-s − 0.937·41-s + 1.37·43-s + 1.34·45-s + 0.437·47-s − 0.274·53-s + 0.404·55-s + 1.56·59-s − 1.40·61-s − 0.977·67-s + 0.712·71-s + 1.75·73-s − 1.35·79-s + 81-s − 0.768·83-s − 1.95·85-s + 0.847·89-s − 0.307·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.73399562032679455937365894379, −7.02028215244095049263424732329, −6.07078470326296157609746681334, −5.45274598721425447175132701183, −4.68576495790058444391416572636, −3.79727198485076033745307334702, −3.26064805881889257533029407709, −2.46158077286027324420675407151, −1.03611344715743476261895002874, 0,
1.03611344715743476261895002874, 2.46158077286027324420675407151, 3.26064805881889257533029407709, 3.79727198485076033745307334702, 4.68576495790058444391416572636, 5.45274598721425447175132701183, 6.07078470326296157609746681334, 7.02028215244095049263424732329, 7.73399562032679455937365894379