L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s + 11-s − 13-s + 3·14-s + 16-s + 2·17-s − 7·19-s + 22-s − 5·25-s − 26-s + 3·28-s + 5·29-s − 2·31-s + 32-s + 2·34-s + 7·37-s − 7·38-s + 6·41-s + 4·43-s + 44-s − 7·47-s + 2·49-s − 5·50-s − 52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s + 0.301·11-s − 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.485·17-s − 1.60·19-s + 0.213·22-s − 25-s − 0.196·26-s + 0.566·28-s + 0.928·29-s − 0.359·31-s + 0.176·32-s + 0.342·34-s + 1.15·37-s − 1.13·38-s + 0.937·41-s + 0.609·43-s + 0.150·44-s − 1.02·47-s + 2/7·49-s − 0.707·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104778 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104778 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5821 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−14.09312185015087, −13.45335248877983, −12.99340536886534, −12.46473297956638, −12.08317425100965, −11.54387130707950, −10.99368254304971, −10.84248392741806, −10.05831478849016, −9.603335511705895, −8.979428698871141, −8.290373797600512, −7.920114150350977, −7.594369545406100, −6.681940446698300, −6.418438533753980, −5.751508773514294, −5.232188501115732, −4.649862372962007, −4.175031605267905, −3.812454631288173, −2.829010992575596, −2.381760479421779, −1.688958844591453, −1.125136056477181, 0,
1.125136056477181, 1.688958844591453, 2.381760479421779, 2.829010992575596, 3.812454631288173, 4.175031605267905, 4.649862372962007, 5.232188501115732, 5.751508773514294, 6.418438533753980, 6.681940446698300, 7.594369545406100, 7.920114150350977, 8.290373797600512, 8.979428698871141, 9.603335511705895, 10.05831478849016, 10.84248392741806, 10.99368254304971, 11.54387130707950, 12.08317425100965, 12.46473297956638, 12.99340536886534, 13.45335248877983, 14.09312185015087