| L(s) = 1 | + 1.30i·2-s + 0.302·4-s + 0.697i·7-s + 3i·8-s + 11-s − 5i·13-s − 0.908·14-s − 3.30·16-s − 6.90i·17-s + 19-s + 1.30i·22-s − 7.30i·23-s + 6.51·26-s + 0.211i·28-s + 0.908·29-s + ⋯ |
| L(s) = 1 | + 0.921i·2-s + 0.151·4-s + 0.263i·7-s + 1.06i·8-s + 0.301·11-s − 1.38i·13-s − 0.242·14-s − 0.825·16-s − 1.67i·17-s + 0.229·19-s + 0.277i·22-s − 1.52i·23-s + 1.27·26-s + 0.0398i·28-s + 0.168·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.077990967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.077990967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.30iT - 2T^{2} \) |
| 7 | \( 1 - 0.697iT - 7T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + 6.90iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 7.30iT - 23T^{2} \) |
| 29 | \( 1 - 0.908T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.39iT - 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 7.21iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 1.30iT - 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 2.60T + 71T^{2} \) |
| 73 | \( 1 - 7.90iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 3.51iT - 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 - 15.3iT - 97T^{2} \) |
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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
−8.753358512210860752390078616833, −8.056697991145490535060495706608, −7.46454422468484014944144736031, −6.64307360393171827411811335032, −6.03037139184096502562985732536, −5.18446376556013042217892433579, −4.55512676959662822987975176086, −3.01528561150048762647909535702, −2.47917784652485348164168697410, −0.75548626732105279576406476618,
1.22905434695490023182364998205, 1.92363824069532620456156983266, 3.06595144756460011219153849576, 3.94520093433371174121114308060, 4.51071127207078746959372477463, 5.96249559579041103814494433760, 6.48419273502534387988851264874, 7.34006115716354732797814891892, 8.130174348523336357948911952355, 9.208461191255520852478701743298
