| L(s) = 1 | − 20·13-s + 48·25-s + 48·37-s − 98·49-s − 240·61-s + 192·73-s − 288·97-s − 364·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.53·13-s + 1.91·25-s + 1.29·37-s − 2·49-s − 3.93·61-s + 2.63·73-s − 2.96·97-s − 3.33·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.224·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9119958306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9119958306\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 480 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1680 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1440 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 5040 T^{2} + p^{4} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12480 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{2} \) |
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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
−9.270542100350163152045461730676, −8.540367627264066963812628938294, −8.233438140587253853821981784961, −7.84813023267881509331664209672, −7.54221127866029483346268421478, −7.01439924277547238011746504690, −6.81310507857374958286761146438, −6.25637194517092908262329306193, −6.04527932824353888194720510008, −5.29665539387993957677737498042, −5.03752340420401594624228253318, −4.62783806926802644101261378461, −4.40084802629761769561590640970, −3.65359016581579352616013211268, −3.17896638730828637839780350686, −2.63081903704421381613794788898, −2.48954405627249435765450686840, −1.56324985060688860019196027432, −1.16093078866664605686047485253, −0.24240646998371592910354209018,
0.24240646998371592910354209018, 1.16093078866664605686047485253, 1.56324985060688860019196027432, 2.48954405627249435765450686840, 2.63081903704421381613794788898, 3.17896638730828637839780350686, 3.65359016581579352616013211268, 4.40084802629761769561590640970, 4.62783806926802644101261378461, 5.03752340420401594624228253318, 5.29665539387993957677737498042, 6.04527932824353888194720510008, 6.25637194517092908262329306193, 6.81310507857374958286761146438, 7.01439924277547238011746504690, 7.54221127866029483346268421478, 7.84813023267881509331664209672, 8.233438140587253853821981784961, 8.540367627264066963812628938294, 9.270542100350163152045461730676
