L(s) = 1 | − 3-s − 0.462·5-s + 7-s + 9-s − 1.32·11-s + 1.78·13-s + 0.462·15-s − 2.39·17-s + 7.04·19-s − 21-s + 23-s − 4.78·25-s − 27-s − 3.32·29-s − 7.04·31-s + 1.32·33-s − 0.462·35-s − 7.32·37-s − 1.78·39-s + 1.04·41-s + 1.13·43-s − 0.462·45-s + 11.0·47-s + 49-s + 2.39·51-s + 0.591·53-s + 0.612·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.206·5-s + 0.377·7-s + 0.333·9-s − 0.399·11-s + 0.495·13-s + 0.119·15-s − 0.581·17-s + 1.61·19-s − 0.218·21-s + 0.208·23-s − 0.957·25-s − 0.192·27-s − 0.617·29-s − 1.26·31-s + 0.230·33-s − 0.0781·35-s − 1.20·37-s − 0.285·39-s + 0.163·41-s + 0.173·43-s − 0.0689·45-s + 1.60·47-s + 0.142·49-s + 0.335·51-s + 0.0812·53-s + 0.0825·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 0.462T + 5T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 - 1.78T + 13T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 19 | \( 1 - 7.04T + 19T^{2} \) |
| 29 | \( 1 + 3.32T + 29T^{2} \) |
| 31 | \( 1 + 7.04T + 31T^{2} \) |
| 37 | \( 1 + 7.32T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 0.591T + 53T^{2} \) |
| 59 | \( 1 - 6.46T + 59T^{2} \) |
| 61 | \( 1 - 0.860T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 2.73T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 7.37T + 89T^{2} \) |
| 97 | \( 1 + 8.11T + 97T^{2} \) |
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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
−7.26552733976399072432180783877, −7.13375951367453045568039632272, −5.74717356347840943021861397705, −5.68139952668086965078713808687, −4.74510633141001600890146618113, −3.96435730573117127229746463035, −3.23732779323222932598160179286, −2.11439243022250460751533825598, −1.21079396516726511565422059844, 0,
1.21079396516726511565422059844, 2.11439243022250460751533825598, 3.23732779323222932598160179286, 3.96435730573117127229746463035, 4.74510633141001600890146618113, 5.68139952668086965078713808687, 5.74717356347840943021861397705, 7.13375951367453045568039632272, 7.26552733976399072432180783877