| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 5·11-s + 13-s + 14-s + 16-s + 7·17-s − 2·19-s + 5·22-s − 9·23-s − 26-s − 28-s + 3·29-s + 3·31-s − 32-s − 7·34-s − 2·37-s + 2·38-s + 6·41-s + 13·43-s − 5·44-s + 9·46-s − 7·47-s + 49-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.50·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.69·17-s − 0.458·19-s + 1.06·22-s − 1.87·23-s − 0.196·26-s − 0.188·28-s + 0.557·29-s + 0.538·31-s − 0.176·32-s − 1.20·34-s − 0.328·37-s + 0.324·38-s + 0.937·41-s + 1.98·43-s − 0.753·44-s + 1.32·46-s − 1.02·47-s + 1/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.56662121701629948619758647896, −6.81105509032922844050489026358, −5.78194706381521192267816405522, −5.72751343386442521609300034210, −4.56540220212085025997319149106, −3.68631559096397763324582979267, −2.84819774140791904032207003278, −2.21031015148533091259166248730, −1.05783287580030785152971735720, 0,
1.05783287580030785152971735720, 2.21031015148533091259166248730, 2.84819774140791904032207003278, 3.68631559096397763324582979267, 4.56540220212085025997319149106, 5.72751343386442521609300034210, 5.78194706381521192267816405522, 6.81105509032922844050489026358, 7.56662121701629948619758647896
