| L(s) = 1 | + 2-s − 2·3-s + 2·4-s + 6·5-s − 2·6-s − 2·7-s + 5·8-s + 3·9-s + 6·10-s − 4·12-s − 2·13-s − 2·14-s − 12·15-s + 5·16-s − 6·17-s + 3·18-s + 4·19-s + 12·20-s + 4·21-s − 2·23-s − 10·24-s + 17·25-s − 2·26-s − 4·27-s − 4·28-s − 2·29-s − 12·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 4-s + 2.68·5-s − 0.816·6-s − 0.755·7-s + 1.76·8-s + 9-s + 1.89·10-s − 1.15·12-s − 0.554·13-s − 0.534·14-s − 3.09·15-s + 5/4·16-s − 1.45·17-s + 0.707·18-s + 0.917·19-s + 2.68·20-s + 0.872·21-s − 0.417·23-s − 2.04·24-s + 17/5·25-s − 0.392·26-s − 0.769·27-s − 0.755·28-s − 0.371·29-s − 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.565855049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.565855049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 194 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 222 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.186304399714621572801474421643, −9.105707724808017798853333694717, −8.289904788476414158229852536349, −7.52872696872587286192296732135, −7.44032184125345161842096835366, −7.02294558392243815964767589542, −6.56164975394797762503175282594, −6.11936701092292613638436984972, −6.09684599860608992804544130679, −5.76094621349309937392180384782, −5.14604717568885430800938250250, −5.07120894113363229560877417380, −4.31400371941154875748496084467, −4.25240172795480316619529293514, −3.33146277758417999291721434520, −2.67762993515336307466655093682, −2.34606150932660084426562243744, −1.83753373310767132335445948860, −1.56929116736556763104516901440, −0.71462530180222431389252271042,
0.71462530180222431389252271042, 1.56929116736556763104516901440, 1.83753373310767132335445948860, 2.34606150932660084426562243744, 2.67762993515336307466655093682, 3.33146277758417999291721434520, 4.25240172795480316619529293514, 4.31400371941154875748496084467, 5.07120894113363229560877417380, 5.14604717568885430800938250250, 5.76094621349309937392180384782, 6.09684599860608992804544130679, 6.11936701092292613638436984972, 6.56164975394797762503175282594, 7.02294558392243815964767589542, 7.44032184125345161842096835366, 7.52872696872587286192296732135, 8.289904788476414158229852536349, 9.105707724808017798853333694717, 9.186304399714621572801474421643
