L(s) = 1 | + 3-s + 2·5-s + 9-s − 4·11-s − 2·13-s + 2·15-s − 6·17-s − 23-s − 25-s + 27-s − 2·29-s + 4·31-s − 4·33-s + 6·37-s − 2·39-s + 6·41-s − 12·43-s + 2·45-s − 12·47-s − 6·51-s + 6·53-s − 8·55-s − 4·59-s + 10·61-s − 4·65-s − 4·67-s − 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.208·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 1.82·43-s + 0.298·45-s − 1.75·47-s − 0.840·51-s + 0.824·53-s − 1.07·55-s − 0.520·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.105911589\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.105911589\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.34513307654598, −13.92220680570510, −13.27410454567362, −13.05412356120601, −12.76475681932050, −11.75333503939918, −11.41605675755948, −10.71546970110879, −10.20284561386587, −9.668873926008237, −9.496531488838848, −8.540973266308804, −8.323058099306256, −7.668780641299802, −7.034615025163774, −6.522729075989260, −5.897709159453227, −5.295774672822322, −4.705143779594285, −4.213980060456619, −3.286975076242293, −2.644663417100188, −2.194841685945922, −1.652132286498091, −0.4496486900189149,
0.4496486900189149, 1.652132286498091, 2.194841685945922, 2.644663417100188, 3.286975076242293, 4.213980060456619, 4.705143779594285, 5.295774672822322, 5.897709159453227, 6.522729075989260, 7.034615025163774, 7.668780641299802, 8.323058099306256, 8.540973266308804, 9.496531488838848, 9.668873926008237, 10.20284561386587, 10.71546970110879, 11.41605675755948, 11.75333503939918, 12.76475681932050, 13.05412356120601, 13.27410454567362, 13.92220680570510, 14.34513307654598