| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s − 2·9-s − 12-s + 13-s − 14-s + 16-s + 3·17-s + 2·18-s + 19-s − 21-s − 3·23-s + 24-s − 26-s + 5·27-s + 28-s − 3·29-s + 2·31-s − 32-s − 3·34-s − 2·36-s + 10·37-s − 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.229·19-s − 0.218·21-s − 0.625·23-s + 0.204·24-s − 0.196·26-s + 0.962·27-s + 0.188·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s − 0.514·34-s − 1/3·36-s + 1.64·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8821121233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8821121233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.983035167615231513039157588703, −9.303912250997036448752264323167, −8.237637220939037195909198183700, −7.77982432388455298937264149525, −6.57249513174536622080526940747, −5.85301595783600076781492520747, −4.98057051522020806813162729411, −3.61241990046837984333714647492, −2.34024048259330359076586962577, −0.850809003224210732230405050623,
0.850809003224210732230405050623, 2.34024048259330359076586962577, 3.61241990046837984333714647492, 4.98057051522020806813162729411, 5.85301595783600076781492520747, 6.57249513174536622080526940747, 7.77982432388455298937264149525, 8.237637220939037195909198183700, 9.303912250997036448752264323167, 9.983035167615231513039157588703
