| L(s) = 1 | − 108·13-s + 4·19-s + 135·25-s + 402·31-s − 404·37-s − 488·43-s − 1.17e3·61-s − 604·67-s − 892·73-s − 534·79-s + 1.19e3·97-s + 3.32e3·103-s + 716·109-s − 2.27e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.35e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2.30·13-s + 0.0482·19-s + 1.07·25-s + 2.32·31-s − 1.79·37-s − 1.73·43-s − 2.46·61-s − 1.10·67-s − 1.43·73-s − 0.760·79-s + 1.24·97-s + 3.18·103-s + 0.629·109-s − 1.71·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.98·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 27 p T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 207 p T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 8286 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 14166 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 39153 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 201 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 202 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 83918 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 244 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 152206 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 121511 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 40773 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 588 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 302 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 691182 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 446 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 267 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 616509 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1396078 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 595 T + p^{3} T^{2} )^{2} \) |
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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
−8.620203649736290657175755995650, −8.586740318124698996292291547351, −7.76552779815951799262175748521, −7.58966804006775143692945830195, −7.19126337671388268557212209084, −6.82564391049204036505926314179, −6.34591529730909993089639037558, −6.02518572393851410653346400632, −5.37574373103573861283061926611, −4.86185259813450380365846914110, −4.65766915532001744102362207869, −4.52043212109484995073628028136, −3.46080834439672071937894281912, −3.21409310645221095313596190477, −2.66585313975709949329502572211, −2.26560523597228159067365366374, −1.58507557891861821758582509406, −1.02484160624818964312765150737, 0, 0,
1.02484160624818964312765150737, 1.58507557891861821758582509406, 2.26560523597228159067365366374, 2.66585313975709949329502572211, 3.21409310645221095313596190477, 3.46080834439672071937894281912, 4.52043212109484995073628028136, 4.65766915532001744102362207869, 4.86185259813450380365846914110, 5.37574373103573861283061926611, 6.02518572393851410653346400632, 6.34591529730909993089639037558, 6.82564391049204036505926314179, 7.19126337671388268557212209084, 7.58966804006775143692945830195, 7.76552779815951799262175748521, 8.586740318124698996292291547351, 8.620203649736290657175755995650
