| L(s) = 1 | + 1.41·2-s − 1.41·3-s − 2.00·6-s − 7-s − 2.82·8-s − 0.999·9-s + 4.24·11-s + 13-s − 1.41·14-s − 4.00·16-s − 1.41·17-s − 1.41·18-s − 7.24·19-s + 1.41·21-s + 6·22-s + 5.82·23-s + 4·24-s + 1.41·26-s + 5.65·27-s + 0.171·29-s + 3.24·31-s − 6·33-s − 2.00·34-s − 2.24·37-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.816·3-s − 0.816·6-s − 0.377·7-s − 0.999·8-s − 0.333·9-s + 1.27·11-s + 0.277·13-s − 0.377·14-s − 1.00·16-s − 0.342·17-s − 0.333·18-s − 1.66·19-s + 0.308·21-s + 1.27·22-s + 1.21·23-s + 0.816·24-s + 0.277·26-s + 1.08·27-s + 0.0318·29-s + 0.582·31-s − 1.04·33-s − 0.342·34-s − 0.368·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.586194494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586194494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 - 5.82T + 23T^{2} \) |
| 29 | \( 1 - 0.171T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 + 2.24T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 - 0.171T + 53T^{2} \) |
| 59 | \( 1 - 0.343T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 1.58T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.910406033472334021761878303607, −8.518212930767048461625161348965, −7.00441240009104713526450222242, −6.33025607128642050639931042922, −5.97047985237160563130376249014, −4.96643189759817778648104500327, −4.29116339999626558377485545147, −3.51991001446845556787659807396, −2.44134627710096431805781094842, −0.72276881933892663336381580117,
0.72276881933892663336381580117, 2.44134627710096431805781094842, 3.51991001446845556787659807396, 4.29116339999626558377485545147, 4.96643189759817778648104500327, 5.97047985237160563130376249014, 6.33025607128642050639931042922, 7.00441240009104713526450222242, 8.518212930767048461625161348965, 8.910406033472334021761878303607
