L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 6·17-s − 4·19-s + 23-s − 25-s + 27-s + 2·29-s − 8·31-s − 4·33-s − 6·37-s − 2·39-s + 6·41-s − 4·43-s − 2·45-s − 8·47-s + 6·51-s − 6·53-s + 8·55-s − 4·57-s − 4·59-s − 10·61-s + 4·65-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 − 0.917·19-s + 0.208·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s + 0.840·51-s − 0.824·53-s + 1.07·55-s − 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.09582682894599, −12.71358711818493, −12.33188518074709, −12.02630842490542, −11.23537626942775, −10.90141583396037, −10.46011994784565, −9.858840123489355, −9.579271739844344, −8.897701293041354, −8.358957202089758, −7.983960324239386, −7.559668205481556, −7.336384579756811, −6.576732147417778, −6.031492985203461, −5.276093278762327, −5.015868547175418, −4.389622893252406, −3.746684374737474, −3.292277978880189, −2.906150695845250, −2.099456955283116, −1.674376091227832, −0.6412142100979749, 0,
0.6412142100979749, 1.674376091227832, 2.099456955283116, 2.906150695845250, 3.292277978880189, 3.746684374737474, 4.389622893252406, 5.015868547175418, 5.276093278762327, 6.031492985203461, 6.576732147417778, 7.336384579756811, 7.559668205481556, 7.983960324239386, 8.358957202089758, 8.897701293041354, 9.579271739844344, 9.858840123489355, 10.46011994784565, 10.90141583396037, 11.23537626942775, 12.02630842490542, 12.33188518074709, 12.71358711818493, 13.09582682894599