L(s) = 1 | + 2-s − 2·3-s − 4-s − 5-s − 2·6-s − 3·8-s + 9-s − 10-s + 2·12-s − 13-s + 2·15-s − 16-s + 5·17-s + 18-s − 6·19-s + 20-s + 2·23-s + 6·24-s − 4·25-s − 26-s + 4·27-s + 9·29-s + 2·30-s + 2·31-s + 5·32-s + 5·34-s − 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s − 0.277·13-s + 0.516·15-s − 1/4·16-s + 1.21·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s + 0.417·23-s + 1.22·24-s − 4/5·25-s − 0.196·26-s + 0.769·27-s + 1.67·29-s + 0.365·30-s + 0.359·31-s + 0.883·32-s + 0.857·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.72141831989896546635194366239, −6.68207180393814649510509109284, −6.15698773960200564988856732359, −5.51102899061673407091097661731, −4.84360107089508271048548448698, −4.30168430717738937425631180351, −3.46637555773943361722890431508, −2.56684794855820615253441453786, −0.981276970722953942837046262399, 0,
0.981276970722953942837046262399, 2.56684794855820615253441453786, 3.46637555773943361722890431508, 4.30168430717738937425631180351, 4.84360107089508271048548448698, 5.51102899061673407091097661731, 6.15698773960200564988856732359, 6.68207180393814649510509109284, 7.72141831989896546635194366239