L(s) = 1 | + (0.587 − 0.190i)2-s + (−0.587 − 0.809i)3-s + (−0.5 + 0.363i)4-s + (−0.5 − 0.363i)6-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.587 + 0.190i)12-s + (0.951 − 1.30i)17-s + 0.618i·18-s + (0.5 + 0.363i)19-s + (1.53 − 0.5i)23-s + 24-s + (0.951 − 0.309i)27-s + (1.30 + 0.951i)31-s − i·32-s + ⋯ |
L(s) = 1 | + (0.587 − 0.190i)2-s + (−0.587 − 0.809i)3-s + (−0.5 + 0.363i)4-s + (−0.5 − 0.363i)6-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.587 + 0.190i)12-s + (0.951 − 1.30i)17-s + 0.618i·18-s + (0.5 + 0.363i)19-s + (1.53 − 0.5i)23-s + 24-s + (0.951 − 0.309i)27-s + (1.30 + 0.951i)31-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.102964694\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.102964694\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.951i)T^{2} \) |
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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
−9.279563031435206673403227457693, −8.444618466965717562969584996691, −7.69689787751784093167377583774, −6.98413310011167736289564778744, −6.04900858092394957519348180113, −5.09168998750343521327133167260, −4.77605650797697560927659012653, −3.31925384718118450513417416969, −2.64055485185069724613858212341, −1.03380563186559177846250721650,
1.07099924914745953065240555302, 3.10252082115213211019250924275, 3.85348899967981939324967904835, 4.71507514817298473091225046890, 5.36307586871881680929557652099, 6.05235416239574908323667194938, 6.77384230945680924200282368790, 7.983785366183398033327942461543, 8.924965223175094357764289221915, 9.601137917293091790891770369262