L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 4·11-s + 12-s + 16-s + 2·17-s − 18-s − 19-s − 4·22-s − 2·23-s − 24-s − 5·25-s + 27-s − 6·29-s − 6·31-s − 32-s + 4·33-s − 2·34-s + 36-s − 8·37-s + 38-s − 10·41-s − 12·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.852·22-s − 0.417·23-s − 0.204·24-s − 25-s + 0.192·27-s − 1.11·29-s − 1.07·31-s − 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s − 1.31·37-s + 0.162·38-s − 1.56·41-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 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
−7.963827187419789211430327684788, −7.12214234772244724841434492496, −6.63116441718754640169590116970, −5.77047250595078190707289552234, −4.88911068840059762819121883590, −3.63810961775377186651852118635, −3.47720922400606981079861592555, −1.98418461348406802916978160680, −1.56373695223205832372938931343, 0,
1.56373695223205832372938931343, 1.98418461348406802916978160680, 3.47720922400606981079861592555, 3.63810961775377186651852118635, 4.88911068840059762819121883590, 5.77047250595078190707289552234, 6.63116441718754640169590116970, 7.12214234772244724841434492496, 7.963827187419789211430327684788