| L(s) = 1 | − 2·5-s − 2·7-s − 4·17-s + 4·19-s + 10·23-s + 3·25-s + 6·29-s − 12·31-s + 4·35-s − 12·37-s − 10·41-s − 4·43-s + 14·47-s − 5·49-s − 4·53-s + 12·59-s − 6·61-s + 2·67-s + 8·73-s + 4·79-s + 6·83-s + 8·85-s − 14·89-s − 8·95-s + 4·97-s − 4·101-s − 20·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.970·17-s + 0.917·19-s + 2.08·23-s + 3/5·25-s + 1.11·29-s − 2.15·31-s + 0.676·35-s − 1.97·37-s − 1.56·41-s − 0.609·43-s + 2.04·47-s − 5/7·49-s − 0.549·53-s + 1.56·59-s − 0.768·61-s + 0.244·67-s + 0.936·73-s + 0.450·79-s + 0.658·83-s + 0.867·85-s − 1.48·89-s − 0.820·95-s + 0.406·97-s − 0.398·101-s − 1.97·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 65 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 83 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 169 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 131 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.61259979343361470049431124958, −7.48476875566517662747041628670, −7.05670769903213810545746249503, −6.92956651314078003575769993649, −6.48506032629016623094252798351, −6.32061538772238072790485298279, −5.45191405024685199131736079253, −5.37627893894708399244255225938, −4.97423659981177996555444549650, −4.72597201617999733469843463947, −4.09740363913011287335324522534, −3.65939877446870327049414797839, −3.40087577282262531337233081886, −3.22156468883634681010563923872, −2.48013527449490758236571527159, −2.28369580479480489476430218731, −1.32234569409703140399230656624, −1.14532498936158647876673186615, 0, 0,
1.14532498936158647876673186615, 1.32234569409703140399230656624, 2.28369580479480489476430218731, 2.48013527449490758236571527159, 3.22156468883634681010563923872, 3.40087577282262531337233081886, 3.65939877446870327049414797839, 4.09740363913011287335324522534, 4.72597201617999733469843463947, 4.97423659981177996555444549650, 5.37627893894708399244255225938, 5.45191405024685199131736079253, 6.32061538772238072790485298279, 6.48506032629016623094252798351, 6.92956651314078003575769993649, 7.05670769903213810545746249503, 7.48476875566517662747041628670, 7.61259979343361470049431124958
