| L(s) = 1 | − 3-s + 7-s + 9-s + 2·11-s − 4·13-s − 2·17-s − 2·19-s − 21-s − 4·23-s − 27-s − 2·29-s − 6·31-s − 2·33-s + 6·37-s + 4·39-s + 6·41-s + 4·43-s + 49-s + 2·51-s − 8·53-s + 2·57-s − 10·61-s + 63-s + 12·67-s + 4·69-s − 14·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.485·17-s − 0.458·19-s − 0.218·21-s − 0.834·23-s − 0.192·27-s − 0.371·29-s − 1.07·31-s − 0.348·33-s + 0.986·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.280·51-s − 1.09·53-s + 0.264·57-s − 1.28·61-s + 0.125·63-s + 1.46·67-s + 0.481·69-s − 1.66·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.810604651707505488801003583145, −7.77428770344336451802547146753, −7.20551876670137642569822391774, −6.27569807606588166609623696943, −5.59422122445989408562937533648, −4.59732957511765143748787214124, −4.02971708798970593661022578819, −2.61799604285331159391151906487, −1.57070282673663074941226872408, 0,
1.57070282673663074941226872408, 2.61799604285331159391151906487, 4.02971708798970593661022578819, 4.59732957511765143748787214124, 5.59422122445989408562937533648, 6.27569807606588166609623696943, 7.20551876670137642569822391774, 7.77428770344336451802547146753, 8.810604651707505488801003583145
