L(s) = 1 | + 2·5-s − 4·7-s − 3·9-s + 2·11-s − 13-s − 17-s − 2·23-s − 25-s + 8·29-s + 8·31-s − 8·35-s − 6·37-s + 12·41-s − 4·43-s − 6·45-s + 8·47-s + 9·49-s − 6·53-s + 4·55-s + 4·59-s − 8·61-s + 12·63-s − 2·65-s + 8·67-s + 8·71-s + 8·73-s − 8·77-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.51·7-s − 9-s + 0.603·11-s − 0.277·13-s − 0.242·17-s − 0.417·23-s − 1/5·25-s + 1.48·29-s + 1.43·31-s − 1.35·35-s − 0.986·37-s + 1.87·41-s − 0.609·43-s − 0.894·45-s + 1.16·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s + 0.520·59-s − 1.02·61-s + 1.51·63-s − 0.248·65-s + 0.977·67-s + 0.949·71-s + 0.936·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.537990138\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.537990138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.744935143171127189763688092005, −7.904622934007829461372572052755, −6.76050381799637175256214886466, −6.31047563927211481477598067176, −5.82694436705930699890394437913, −4.83916459898418813319356088886, −3.76489209252055257793946503303, −2.90554289030514861919259038388, −2.24018548227623988032532843616, −0.70436545265522477837812829815,
0.70436545265522477837812829815, 2.24018548227623988032532843616, 2.90554289030514861919259038388, 3.76489209252055257793946503303, 4.83916459898418813319356088886, 5.82694436705930699890394437913, 6.31047563927211481477598067176, 6.76050381799637175256214886466, 7.904622934007829461372572052755, 8.744935143171127189763688092005