L(s) = 1 | + 3-s + 5-s + 9-s + 2·11-s + 15-s + 6·17-s + 8·19-s − 23-s + 25-s + 27-s − 8·29-s − 8·31-s + 2·33-s − 2·37-s + 6·41-s + 6·43-s + 45-s − 4·47-s − 7·49-s + 6·51-s − 6·53-s + 2·55-s + 8·57-s + 2·61-s − 2·67-s − 69-s + 14·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.258·15-s + 1.45·17-s + 1.83·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s + 0.348·33-s − 0.328·37-s + 0.937·41-s + 0.914·43-s + 0.149·45-s − 0.583·47-s − 49-s + 0.840·51-s − 0.824·53-s + 0.269·55-s + 1.05·57-s + 0.256·61-s − 0.244·67-s − 0.120·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.810756957\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.810756957\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−9.118759237493293756115638694934, −7.73052462537736614451635947593, −7.66544344625033897958161976044, −6.56418873968097065615493271281, −5.63452783598334061771707104863, −5.07807928988700731659791180838, −3.72310748489463193162600086347, −3.30360531522552296438400266076, −2.05167018221242797682190777843, −1.10373938578605422358061549806,
1.10373938578605422358061549806, 2.05167018221242797682190777843, 3.30360531522552296438400266076, 3.72310748489463193162600086347, 5.07807928988700731659791180838, 5.63452783598334061771707104863, 6.56418873968097065615493271281, 7.66544344625033897958161976044, 7.73052462537736614451635947593, 9.118759237493293756115638694934