L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 2·13-s + 14-s + 16-s − 6·17-s + 8·19-s + 2·26-s − 28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 10·37-s − 8·38-s + 6·41-s + 4·43-s + 49-s − 2·52-s − 6·53-s + 56-s + 6·58-s + 12·59-s − 10·61-s + 4·62-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 1.83·19-s + 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1.64·37-s − 1.29·38-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.277·52-s − 0.824·53-s + 0.133·56-s + 0.787·58-s + 1.56·59-s − 1.28·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.044053307\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.044053307\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.923782644459481778947909429260, −7.73143322454406199184768561933, −7.44924214038627060094492752298, −6.54330061765932504519811413374, −5.78970024404309668434570027328, −4.89780053825845340735931076352, −3.86792820022911429931360386402, −2.87517927529527431994089643143, −2.01094706671842157218913301495, −0.67052284488312117191785422744,
0.67052284488312117191785422744, 2.01094706671842157218913301495, 2.87517927529527431994089643143, 3.86792820022911429931360386402, 4.89780053825845340735931076352, 5.78970024404309668434570027328, 6.54330061765932504519811413374, 7.44924214038627060094492752298, 7.73143322454406199184768561933, 8.923782644459481778947909429260