| L(s) = 1 | − 0.741·2-s − 3-s − 1.45·4-s + 0.741·6-s − 1.03·7-s + 2.55·8-s + 9-s − 0.513·11-s + 1.45·12-s − 3.54·13-s + 0.767·14-s + 1.00·16-s + 1.36·17-s − 0.741·18-s + 0.894·19-s + 1.03·21-s + 0.380·22-s − 5.45·23-s − 2.55·24-s + 2.62·26-s − 27-s + 1.50·28-s − 9.65·29-s + 10.4·31-s − 5.86·32-s + 0.513·33-s − 1.01·34-s + ⋯ |
| L(s) = 1 | − 0.524·2-s − 0.577·3-s − 0.725·4-s + 0.302·6-s − 0.391·7-s + 0.904·8-s + 0.333·9-s − 0.154·11-s + 0.418·12-s − 0.982·13-s + 0.205·14-s + 0.251·16-s + 0.331·17-s − 0.174·18-s + 0.205·19-s + 0.226·21-s + 0.0812·22-s − 1.13·23-s − 0.522·24-s + 0.515·26-s − 0.192·27-s + 0.283·28-s − 1.79·29-s + 1.87·31-s − 1.03·32-s + 0.0894·33-s − 0.173·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5575351034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5575351034\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 0.741T + 2T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 + 0.513T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 - 1.36T + 17T^{2} \) |
| 19 | \( 1 - 0.894T + 19T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 2.19T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 7.65T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 + 8.80T + 67T^{2} \) |
| 71 | \( 1 - 5.00T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 - 7.99T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−9.446074005619675223036941603683, −8.413755444054455082303781135263, −7.76279443205693034579240122492, −6.98530545054495088604583613062, −5.97830057858293646853204432696, −5.16602917735454224098344326586, −4.41974923102374555369838966797, −3.43428873479743862147949364736, −1.99923046024742581479817625031, −0.55625074122095507039006727413,
0.55625074122095507039006727413, 1.99923046024742581479817625031, 3.43428873479743862147949364736, 4.41974923102374555369838966797, 5.16602917735454224098344326586, 5.97830057858293646853204432696, 6.98530545054495088604583613062, 7.76279443205693034579240122492, 8.413755444054455082303781135263, 9.446074005619675223036941603683
