| L(s) = 1 | + 10·3-s + 10·5-s + 21·9-s + 100·15-s − 70·23-s − 175·25-s − 310·27-s − 30·31-s − 530·37-s + 210·45-s − 760·47-s − 270·49-s + 1.02e3·53-s − 42·59-s − 1.17e3·67-s − 700·69-s − 626·71-s − 1.75e3·75-s − 2.78e3·81-s − 370·89-s − 300·93-s + 1.57e3·97-s − 680·103-s − 5.30e3·111-s − 3.75e3·113-s − 700·115-s − 3.25e3·125-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 0.894·5-s + 7/9·9-s + 1.72·15-s − 0.634·23-s − 7/5·25-s − 2.20·27-s − 0.173·31-s − 2.35·37-s + 0.695·45-s − 2.35·47-s − 0.787·49-s + 2.64·53-s − 0.0926·59-s − 2.13·67-s − 1.22·69-s − 1.04·71-s − 2.69·75-s − 3.82·81-s − 0.440·89-s − 0.334·93-s + 1.64·97-s − 0.650·103-s − 4.53·111-s − 3.12·113-s − 0.567·115-s − 2.32·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3748096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3748096 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 5 T + p^{3} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 270 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 50 p T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9410 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3318 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 35 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 7178 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 265 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 127442 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42330 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 380 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 510 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 21 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 412362 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 585 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 313 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 557970 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 611678 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 717590 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 185 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 785 T + p^{3} T^{2} )^{2} \) |
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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
−8.567441689248121786050943649173, −8.406238717581870970665467471004, −7.892567963769392815710824141348, −7.76668842643779232161794302610, −7.05282752871815987031633963380, −6.91188050936427045115020990896, −6.02356267396606478462323700842, −5.99554911633185765370015879054, −5.46210328299267394485167335919, −5.07533305330982728582963241080, −4.38732673657760975024727987202, −3.87293148745335885014558126228, −3.43569738306039917452511319502, −3.23045038660188913462276342097, −2.51122180717434966296845726513, −2.27377461151570966139577951310, −1.69519789516285167485933799656, −1.47249963207834905749924754618, 0, 0,
1.47249963207834905749924754618, 1.69519789516285167485933799656, 2.27377461151570966139577951310, 2.51122180717434966296845726513, 3.23045038660188913462276342097, 3.43569738306039917452511319502, 3.87293148745335885014558126228, 4.38732673657760975024727987202, 5.07533305330982728582963241080, 5.46210328299267394485167335919, 5.99554911633185765370015879054, 6.02356267396606478462323700842, 6.91188050936427045115020990896, 7.05282752871815987031633963380, 7.76668842643779232161794302610, 7.892567963769392815710824141348, 8.406238717581870970665467471004, 8.567441689248121786050943649173
