| L(s) = 1 | + 2·2-s + 3·4-s − 2·7-s + 4·8-s − 2·13-s − 4·14-s + 5·16-s + 2·17-s − 4·19-s − 2·23-s − 4·26-s − 6·28-s − 10·29-s + 2·31-s + 6·32-s + 4·34-s − 8·37-s − 8·38-s + 2·43-s − 4·46-s + 4·47-s + 3·49-s − 6·52-s − 6·53-s − 8·56-s − 20·58-s − 10·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s − 0.554·13-s − 1.06·14-s + 5/4·16-s + 0.485·17-s − 0.917·19-s − 0.417·23-s − 0.784·26-s − 1.13·28-s − 1.85·29-s + 0.359·31-s + 1.06·32-s + 0.685·34-s − 1.31·37-s − 1.29·38-s + 0.304·43-s − 0.589·46-s + 0.583·47-s + 3/7·49-s − 0.832·52-s − 0.824·53-s − 1.06·56-s − 2.62·58-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 77 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 109 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 119 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 145 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 168 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 332 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−7.25235717032043167988560551240, −7.13118123961316428919462994785, −6.70449515477272964278672457636, −6.62373508989685694367545855441, −5.93690824051396671841798556188, −5.77638699855281663981503800813, −5.46158304410155166259654822598, −5.31046390515458387469956022780, −4.53280816433302398377706379506, −4.47342034800037097658573544578, −3.91824062602671063903240934109, −3.84216370159458564912897245354, −3.17362689453215552763616118929, −3.04767942455355005141932825649, −2.40678780384455510814128033190, −2.29079002649162862105070764390, −1.44163522678942335387059079644, −1.39301760508774699725673714588, 0, 0,
1.39301760508774699725673714588, 1.44163522678942335387059079644, 2.29079002649162862105070764390, 2.40678780384455510814128033190, 3.04767942455355005141932825649, 3.17362689453215552763616118929, 3.84216370159458564912897245354, 3.91824062602671063903240934109, 4.47342034800037097658573544578, 4.53280816433302398377706379506, 5.31046390515458387469956022780, 5.46158304410155166259654822598, 5.77638699855281663981503800813, 5.93690824051396671841798556188, 6.62373508989685694367545855441, 6.70449515477272964278672457636, 7.13118123961316428919462994785, 7.25235717032043167988560551240
