| L(s) = 1 | − 3·5-s − 7-s + 6·11-s − 13-s + 3·17-s − 2·19-s + 4·25-s + 6·29-s − 4·31-s + 3·35-s + 7·37-s + 43-s − 3·47-s − 6·49-s − 18·55-s − 6·59-s − 8·61-s + 3·65-s − 14·67-s + 3·71-s + 2·73-s − 6·77-s + 8·79-s + 12·83-s − 9·85-s + 6·89-s + 91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 1.80·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s + 4/5·25-s + 1.11·29-s − 0.718·31-s + 0.507·35-s + 1.15·37-s + 0.152·43-s − 0.437·47-s − 6/7·49-s − 2.42·55-s − 0.781·59-s − 1.02·61-s + 0.372·65-s − 1.71·67-s + 0.356·71-s + 0.234·73-s − 0.683·77-s + 0.900·79-s + 1.31·83-s − 0.976·85-s + 0.635·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.420436029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420436029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.84945114216431970416812906623, −7.25909207276503257739570375205, −6.50651624348626722423445910356, −6.03717365891291117513273597634, −4.81812624433810441947753794628, −4.24183714628839419491149363112, −3.60085571891222062838545931539, −2.97646602709988554834027991523, −1.62932437964747282291652744911, −0.62006701852715344476553743600,
0.62006701852715344476553743600, 1.62932437964747282291652744911, 2.97646602709988554834027991523, 3.60085571891222062838545931539, 4.24183714628839419491149363112, 4.81812624433810441947753794628, 6.03717365891291117513273597634, 6.50651624348626722423445910356, 7.25909207276503257739570375205, 7.84945114216431970416812906623
