| L(s) = 1 | − 2·3-s + 3·7-s + 9-s − 6·11-s − 4·13-s + 3·17-s + 6·19-s − 6·21-s + 4·27-s + 3·29-s + 5·31-s + 12·33-s + 3·37-s + 8·39-s + 9·41-s + 12·43-s − 6·47-s + 2·49-s − 6·51-s + 3·53-s − 12·57-s + 9·59-s − 6·61-s + 3·63-s − 9·67-s − 9·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.13·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.727·17-s + 1.37·19-s − 1.30·21-s + 0.769·27-s + 0.557·29-s + 0.898·31-s + 2.08·33-s + 0.493·37-s + 1.28·39-s + 1.40·41-s + 1.82·43-s − 0.875·47-s + 2/7·49-s − 0.840·51-s + 0.412·53-s − 1.58·57-s + 1.17·59-s − 0.768·61-s + 0.377·63-s − 1.09·67-s − 1.06·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.05392731426527, −12.65156353823360, −12.25603403002376, −11.75029277631397, −11.39145686969355, −11.05229456373886, −10.35193653410797, −10.20792378111684, −9.714481696685068, −8.993236205532612, −8.399461991562158, −7.811567008797254, −7.523960966444346, −7.288861165548434, −6.365164236195799, −5.746734683279592, −5.550861974490278, −4.939695656160431, −4.752910372217913, −4.191069901740983, −3.077486329097290, −2.746856458827246, −2.214390302360569, −1.217993862186313, −0.7908601968261843, 0,
0.7908601968261843, 1.217993862186313, 2.214390302360569, 2.746856458827246, 3.077486329097290, 4.191069901740983, 4.752910372217913, 4.939695656160431, 5.550861974490278, 5.746734683279592, 6.365164236195799, 7.288861165548434, 7.523960966444346, 7.811567008797254, 8.399461991562158, 8.993236205532612, 9.714481696685068, 10.20792378111684, 10.35193653410797, 11.05229456373886, 11.39145686969355, 11.75029277631397, 12.25603403002376, 12.65156353823360, 13.05392731426527
