| L(s) = 1 | + 2.58·2-s − 3-s + 4.65·4-s − 2.58·6-s + 7-s + 6.86·8-s + 9-s + 11-s − 4.65·12-s + 4.47·13-s + 2.58·14-s + 8.39·16-s + 2.67·17-s + 2.58·18-s − 0.383·19-s − 21-s + 2.58·22-s − 0.698·23-s − 6.86·24-s + 11.5·26-s − 27-s + 4.65·28-s − 4.91·29-s + 7.97·31-s + 7.92·32-s − 33-s + 6.91·34-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.577·3-s + 2.32·4-s − 1.05·6-s + 0.377·7-s + 2.42·8-s + 0.333·9-s + 0.301·11-s − 1.34·12-s + 1.24·13-s + 0.689·14-s + 2.09·16-s + 0.649·17-s + 0.608·18-s − 0.0879·19-s − 0.218·21-s + 0.550·22-s − 0.145·23-s − 1.40·24-s + 2.26·26-s − 0.192·27-s + 0.880·28-s − 0.912·29-s + 1.43·31-s + 1.40·32-s − 0.174·33-s + 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.479175050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.479175050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 + 0.383T + 19T^{2} \) |
| 23 | \( 1 + 0.698T + 23T^{2} \) |
| 29 | \( 1 + 4.91T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 + 6.06T + 37T^{2} \) |
| 41 | \( 1 + 2.14T + 41T^{2} \) |
| 43 | \( 1 + 1.16T + 43T^{2} \) |
| 47 | \( 1 - 5.59T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 - 2.60T + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 - 7.37T + 67T^{2} \) |
| 71 | \( 1 - 0.538T + 71T^{2} \) |
| 73 | \( 1 + 6.84T + 73T^{2} \) |
| 79 | \( 1 - 3.87T + 79T^{2} \) |
| 83 | \( 1 - 0.126T + 83T^{2} \) |
| 89 | \( 1 + 7.75T + 89T^{2} \) |
| 97 | \( 1 + 0.987T + 97T^{2} \) |
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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
−7.88273133566297238801891856687, −6.92872225173231921945220763612, −6.51705108930996919214601337178, −5.66820424919209925955647080080, −5.36708057693379310770567920670, −4.42731350339880395128081174640, −3.86155093879962294363975880249, −3.16842397158652949383810457988, −2.06254939906838296570261477527, −1.15448330620273051632271851327,
1.15448330620273051632271851327, 2.06254939906838296570261477527, 3.16842397158652949383810457988, 3.86155093879962294363975880249, 4.42731350339880395128081174640, 5.36708057693379310770567920670, 5.66820424919209925955647080080, 6.51705108930996919214601337178, 6.92872225173231921945220763612, 7.88273133566297238801891856687
