| L(s) = 1 | − 2·3-s + 5-s + 9-s − 2·13-s − 2·15-s + 6·17-s − 4·19-s − 6·23-s + 25-s + 4·27-s + 6·29-s − 4·31-s + 2·37-s + 4·39-s − 6·41-s + 10·43-s + 45-s − 6·47-s − 12·51-s − 6·53-s + 8·57-s + 12·59-s − 2·61-s − 2·65-s − 2·67-s + 12·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 1.45·17-s − 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 1.52·43-s + 0.149·45-s − 0.875·47-s − 1.68·51-s − 0.824·53-s + 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.244·67-s + 1.44·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067513462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067513462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.371569178851438535097195720359, −7.72417967231205206440371091917, −6.75938606427748610323373549762, −6.16943207782230729958538690385, −5.54097686482462236146375699951, −4.92936609872116028004359556395, −4.02348966113043943975885714437, −2.91432418996240576588953142404, −1.84751760735400887993660553283, −0.62674328168244276773408798542,
0.62674328168244276773408798542, 1.84751760735400887993660553283, 2.91432418996240576588953142404, 4.02348966113043943975885714437, 4.92936609872116028004359556395, 5.54097686482462236146375699951, 6.16943207782230729958538690385, 6.75938606427748610323373549762, 7.72417967231205206440371091917, 8.371569178851438535097195720359
