| L(s) = 1 | − 7-s − 3·9-s + 4·11-s − 3·13-s + 17-s − 23-s + 4·31-s + 11·37-s − 10·41-s + 2·43-s + 11·47-s + 49-s + 53-s − 8·59-s + 8·61-s + 3·63-s + 4·71-s + 4·73-s − 4·77-s + 11·79-s + 9·81-s + 13·83-s + 89-s + 3·91-s + 7·97-s − 12·99-s + 101-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.20·11-s − 0.832·13-s + 0.242·17-s − 0.208·23-s + 0.718·31-s + 1.80·37-s − 1.56·41-s + 0.304·43-s + 1.60·47-s + 1/7·49-s + 0.137·53-s − 1.04·59-s + 1.02·61-s + 0.377·63-s + 0.474·71-s + 0.468·73-s − 0.455·77-s + 1.23·79-s + 81-s + 1.42·83-s + 0.105·89-s + 0.314·91-s + 0.710·97-s − 1.20·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.476242677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.476242677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.67084590332754, −12.28701593021808, −11.89167803547039, −11.58289819075446, −11.03568176666705, −10.51661592552647, −9.997331419586820, −9.473256564640899, −9.210743227716515, −8.701490075765773, −8.105608151603749, −7.748565011066807, −7.134755247252410, −6.501686403325032, −6.327863115116814, −5.675543947436784, −5.212002756279756, −4.596456603298286, −4.064656800411036, −3.526687399771857, −2.974149496286716, −2.423424658998218, −1.900846140916541, −0.9499764196182108, −0.5098217458681534,
0.5098217458681534, 0.9499764196182108, 1.900846140916541, 2.423424658998218, 2.974149496286716, 3.526687399771857, 4.064656800411036, 4.596456603298286, 5.212002756279756, 5.675543947436784, 6.327863115116814, 6.501686403325032, 7.134755247252410, 7.748565011066807, 8.105608151603749, 8.701490075765773, 9.210743227716515, 9.473256564640899, 9.997331419586820, 10.51661592552647, 11.03568176666705, 11.58289819075446, 11.89167803547039, 12.28701593021808, 12.67084590332754
