| L(s) = 1 | + 1.90·2-s + 1.62·4-s − 7-s − 0.719·8-s − 2·11-s − 6.42·13-s − 1.90·14-s − 4.61·16-s − 4.42·17-s − 2.42·19-s − 3.80·22-s − 1.37·23-s − 12.2·26-s − 1.62·28-s − 0.755·29-s + 5.18·31-s − 7.34·32-s − 8.42·34-s + 7.61·37-s − 4.62·38-s + 8.23·41-s + 10.1·43-s − 3.24·44-s − 2.62·46-s − 2.75·47-s + 49-s − 10.4·52-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.811·4-s − 0.377·7-s − 0.254·8-s − 0.603·11-s − 1.78·13-s − 0.508·14-s − 1.15·16-s − 1.07·17-s − 0.557·19-s − 0.811·22-s − 0.287·23-s − 2.39·26-s − 0.306·28-s − 0.140·29-s + 0.931·31-s − 1.29·32-s − 1.44·34-s + 1.25·37-s − 0.749·38-s + 1.28·41-s + 1.54·43-s − 0.489·44-s − 0.386·46-s − 0.401·47-s + 0.142·49-s − 1.44·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 6.42T + 13T^{2} \) |
| 17 | \( 1 + 4.42T + 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 - 8.23T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 2.75T + 47T^{2} \) |
| 53 | \( 1 + 9.18T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 2.75T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 1.57T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−9.205195271999737316687616722725, −8.051732500588590234607731069979, −7.19673640898004520226375015103, −6.36155038002573696295036504564, −5.63889015815739879273075395818, −4.61973301882331014432301187037, −4.28203287705132824613363045401, −2.85528554124399011331205303557, −2.38236433365193009664218653670, 0,
2.38236433365193009664218653670, 2.85528554124399011331205303557, 4.28203287705132824613363045401, 4.61973301882331014432301187037, 5.63889015815739879273075395818, 6.36155038002573696295036504564, 7.19673640898004520226375015103, 8.051732500588590234607731069979, 9.205195271999737316687616722725
