| L(s) = 1 | + 2-s − 0.803·3-s + 4-s − 0.712·5-s − 0.803·6-s + 8-s − 2.35·9-s − 0.712·10-s + 2.61·11-s − 0.803·12-s − 1.95·13-s + 0.572·15-s + 16-s + 3.59·17-s − 2.35·18-s − 7.06·19-s − 0.712·20-s + 2.61·22-s + 3.24·23-s − 0.803·24-s − 4.49·25-s − 1.95·26-s + 4.30·27-s + 29-s + 0.572·30-s − 8.32·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.464·3-s + 0.5·4-s − 0.318·5-s − 0.328·6-s + 0.353·8-s − 0.784·9-s − 0.225·10-s + 0.789·11-s − 0.232·12-s − 0.542·13-s + 0.147·15-s + 0.250·16-s + 0.870·17-s − 0.554·18-s − 1.62·19-s − 0.159·20-s + 0.558·22-s + 0.677·23-s − 0.164·24-s − 0.898·25-s − 0.383·26-s + 0.828·27-s + 0.185·29-s + 0.104·30-s − 1.49·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.803T + 3T^{2} \) |
| 5 | \( 1 + 0.712T + 5T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 + 1.95T + 13T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 + 7.06T + 19T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 31 | \( 1 + 8.32T + 31T^{2} \) |
| 37 | \( 1 - 0.815T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 - 3.35T + 43T^{2} \) |
| 47 | \( 1 + 0.930T + 47T^{2} \) |
| 53 | \( 1 + 5.55T + 53T^{2} \) |
| 59 | \( 1 + 8.89T + 59T^{2} \) |
| 61 | \( 1 + 1.48T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 - 6.03T + 71T^{2} \) |
| 73 | \( 1 - 7.83T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 3.96T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.347011732709536555174796223589, −7.51395241856451813967760559229, −6.70851577894014604712194845579, −6.01879108177547814247754567903, −5.34869432723042046199494224802, −4.48232370437648394249605791105, −3.69777741980674988200530810567, −2.79519151764321445708574017486, −1.63151978422710404275866692132, 0,
1.63151978422710404275866692132, 2.79519151764321445708574017486, 3.69777741980674988200530810567, 4.48232370437648394249605791105, 5.34869432723042046199494224802, 6.01879108177547814247754567903, 6.70851577894014604712194845579, 7.51395241856451813967760559229, 8.347011732709536555174796223589
