L(s) = 1 | − 5-s + 4·11-s − 6·13-s + 6·17-s − 19-s − 8·23-s + 25-s + 2·29-s + 2·37-s − 2·41-s + 4·43-s + 8·47-s − 7·49-s + 6·53-s − 4·55-s + 4·59-s − 2·61-s + 6·65-s + 8·67-s − 8·71-s + 2·73-s − 8·79-s − 4·83-s − 6·85-s + 14·89-s + 95-s + 14·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.229·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 49-s + 0.824·53-s − 0.539·55-s + 0.520·59-s − 0.256·61-s + 0.744·65-s + 0.977·67-s − 0.949·71-s + 0.234·73-s − 0.900·79-s − 0.439·83-s − 0.650·85-s + 1.48·89-s + 0.102·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.643181301\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.643181301\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77182518400080572382455334941, −7.44635733656002322256581917680, −6.59405268264748259181519899604, −5.89076346527159464421717071523, −5.09672192576351087779899052654, −4.27386992884279422456093887883, −3.70426243159380732570972182763, −2.74802343247202258761722407286, −1.81978843055366923229379333876, −0.65533742603902575573928330110,
0.65533742603902575573928330110, 1.81978843055366923229379333876, 2.74802343247202258761722407286, 3.70426243159380732570972182763, 4.27386992884279422456093887883, 5.09672192576351087779899052654, 5.89076346527159464421717071523, 6.59405268264748259181519899604, 7.44635733656002322256581917680, 7.77182518400080572382455334941