| L(s) = 1 | − 3-s + 4·5-s − 2·7-s + 9-s − 4·15-s + 17-s + 2·21-s + 6·23-s + 11·25-s − 27-s + 2·29-s − 4·31-s − 8·35-s + 2·37-s + 6·41-s − 4·43-s + 4·45-s + 6·47-s − 3·49-s − 51-s + 8·53-s − 8·59-s + 8·61-s − 2·63-s + 4·67-s − 6·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.03·15-s + 0.242·17-s + 0.436·21-s + 1.25·23-s + 11/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 1.35·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.596·45-s + 0.875·47-s − 3/7·49-s − 0.140·51-s + 1.09·53-s − 1.04·59-s + 1.02·61-s − 0.251·63-s + 0.488·67-s − 0.722·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.077774124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.077774124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.70034656755983, −13.09198903722023, −12.94111463613882, −12.51733222787569, −11.84314809343380, −11.15727645614139, −10.80464935314715, −10.21776655620346, −9.806505910740757, −9.522379483301498, −8.857150393008056, −8.568083658009400, −7.498977533254467, −7.073310416968507, −6.585401677888379, −6.013041442297335, −5.742630773492164, −5.160340901788321, −4.680463967850363, −3.871064580770011, −3.083486077949645, −2.626393812352449, −1.931037105992673, −1.246407956125393, −0.6045232273895422,
0.6045232273895422, 1.246407956125393, 1.931037105992673, 2.626393812352449, 3.083486077949645, 3.871064580770011, 4.680463967850363, 5.160340901788321, 5.742630773492164, 6.013041442297335, 6.585401677888379, 7.073310416968507, 7.498977533254467, 8.568083658009400, 8.857150393008056, 9.522379483301498, 9.806505910740757, 10.21776655620346, 10.80464935314715, 11.15727645614139, 11.84314809343380, 12.51733222787569, 12.94111463613882, 13.09198903722023, 13.70034656755983
