| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 11-s + 6·13-s + 4·14-s + 16-s + 2·17-s + 4·19-s + 22-s + 4·23-s + 6·26-s + 4·28-s − 6·29-s + 32-s + 2·34-s − 6·37-s + 4·38-s + 6·41-s − 4·43-s + 44-s + 4·46-s − 12·47-s + 9·49-s + 6·52-s + 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 0.301·11-s + 1.66·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.213·22-s + 0.834·23-s + 1.17·26-s + 0.755·28-s − 1.11·29-s + 0.176·32-s + 0.342·34-s − 0.986·37-s + 0.648·38-s + 0.937·41-s − 0.609·43-s + 0.150·44-s + 0.589·46-s − 1.75·47-s + 9/7·49-s + 0.832·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.532030251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.532030251\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.083346957159341864647544158303, −7.60733258179249456185408639578, −6.72859349416794076405626890294, −5.91180395613593790047875267063, −5.26497995912959248632200600675, −4.64260297693215751255366534917, −3.73808402977774882375523442801, −3.11888676425783266508989971268, −1.73607899388152096241559479071, −1.24048608768025684580275578535,
1.24048608768025684580275578535, 1.73607899388152096241559479071, 3.11888676425783266508989971268, 3.73808402977774882375523442801, 4.64260297693215751255366534917, 5.26497995912959248632200600675, 5.91180395613593790047875267063, 6.72859349416794076405626890294, 7.60733258179249456185408639578, 8.083346957159341864647544158303
