L(s) = 1 | + 2-s − 1.41·3-s + 4-s − 1.41·6-s + 8-s − 0.999·9-s − 2·11-s − 1.41·12-s + 16-s + 7.07·17-s − 0.999·18-s − 1.41·19-s − 2·22-s − 1.41·24-s − 5·25-s + 5.65·27-s − 6·29-s + 32-s + 2.82·33-s + 7.07·34-s − 0.999·36-s − 2·37-s − 1.41·38-s + 7.07·41-s − 6·43-s − 2·44-s + 2.82·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.816·3-s + 0.5·4-s − 0.577·6-s + 0.353·8-s − 0.333·9-s − 0.603·11-s − 0.408·12-s + 0.250·16-s + 1.71·17-s − 0.235·18-s − 0.324·19-s − 0.426·22-s − 0.288·24-s − 25-s + 1.08·27-s − 1.11·29-s + 0.176·32-s + 0.492·33-s + 1.21·34-s − 0.166·36-s − 0.328·37-s − 0.229·38-s + 1.10·41-s − 0.914·43-s − 0.301·44-s + 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5194 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5194 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 53 | \( 1 - T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−7.73484004825153356705822525206, −7.03881478550339280851073322932, −6.07703208479603311896104854643, −5.63487520072029528124312462262, −5.16444369534845033467509812323, −4.19528812608915231434809424783, −3.37266344297077226701821464763, −2.54472466158891593617956756647, −1.37233738347896103159440792186, 0,
1.37233738347896103159440792186, 2.54472466158891593617956756647, 3.37266344297077226701821464763, 4.19528812608915231434809424783, 5.16444369534845033467509812323, 5.63487520072029528124312462262, 6.07703208479603311896104854643, 7.03881478550339280851073322932, 7.73484004825153356705822525206