| L(s) = 1 | − 4-s − 2·9-s − 2·11-s + 16-s − 4·19-s + 2·29-s − 2·31-s + 2·36-s + 2·44-s + 49-s + 2·59-s − 2·61-s − 64-s + 4·71-s + 4·76-s + 3·81-s + 4·99-s − 2·101-s − 2·116-s + 121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s − 2·9-s − 2·11-s + 16-s − 4·19-s + 2·29-s − 2·31-s + 2·36-s + 2·44-s + 49-s + 2·59-s − 2·61-s − 64-s + 4·71-s + 4·76-s + 3·81-s + 4·99-s − 2·101-s − 2·116-s + 121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2360238743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2360238743\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$ | \( ( 1 - T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.14939268861253547092188133809, −9.590482463971844013884246458732, −9.072778385182867909067229446814, −8.606835227133804393802160980483, −8.566172911610992649824031318143, −8.083584360028839262414322171855, −8.023252877585771856752682921398, −7.28180024083949205942801585624, −6.46400659371258947028069054052, −6.44802459365064090393557607418, −5.69118940197869931846961230451, −5.53816751989610163750102876166, −4.84044405615061369955862994604, −4.81022527178156155475086427574, −3.81166120839182335378966351532, −3.79819652603281739786361280169, −2.66793167606141221108087714194, −2.60243145073435821673813454710, −2.02299184463387530260001747009, −0.38948178818093216244514842656,
0.38948178818093216244514842656, 2.02299184463387530260001747009, 2.60243145073435821673813454710, 2.66793167606141221108087714194, 3.79819652603281739786361280169, 3.81166120839182335378966351532, 4.81022527178156155475086427574, 4.84044405615061369955862994604, 5.53816751989610163750102876166, 5.69118940197869931846961230451, 6.44802459365064090393557607418, 6.46400659371258947028069054052, 7.28180024083949205942801585624, 8.023252877585771856752682921398, 8.083584360028839262414322171855, 8.566172911610992649824031318143, 8.606835227133804393802160980483, 9.072778385182867909067229446814, 9.590482463971844013884246458732, 10.14939268861253547092188133809
