| L(s) = 1 | + (0.459 − 0.413i)2-s + (−0.0646 + 0.614i)4-s + (0.743 + 0.669i)5-s + (−0.309 + 0.535i)7-s + (0.587 + 0.809i)8-s + 0.618·10-s + (−0.743 + 0.669i)11-s + (0.0794 + 0.373i)14-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.459 + 0.413i)20-s + (−0.0646 + 0.614i)22-s + (−0.207 − 0.978i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.224i)28-s + (0.658 + 1.47i)29-s + ⋯ |
| L(s) = 1 | + (0.459 − 0.413i)2-s + (−0.0646 + 0.614i)4-s + (0.743 + 0.669i)5-s + (−0.309 + 0.535i)7-s + (0.587 + 0.809i)8-s + 0.618·10-s + (−0.743 + 0.669i)11-s + (0.0794 + 0.373i)14-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.459 + 0.413i)20-s + (−0.0646 + 0.614i)22-s + (−0.207 − 0.978i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.224i)28-s + (0.658 + 1.47i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.513869289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513869289\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.743 - 0.669i)T \) |
| good | 2 | \( 1 + (-0.459 + 0.413i)T + (0.104 - 0.994i)T^{2} \) |
| 7 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.743 - 0.669i)T + (0.104 - 0.994i)T^{2} \) |
| 13 | \( 1 + (-0.104 - 0.994i)T^{2} \) |
| 17 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2} \) |
| 29 | \( 1 + (-0.658 - 1.47i)T + (-0.669 + 0.743i)T^{2} \) |
| 31 | \( 1 + (0.669 + 0.743i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.658 + 1.47i)T + (-0.669 + 0.743i)T^{2} \) |
| 53 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 61 | \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 67 | \( 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2} \) |
| 71 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2} \) |
| 83 | \( 1 + (-0.614 + 0.0646i)T + (0.978 - 0.207i)T^{2} \) |
| 89 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.476945618234578591639502369491, −8.807886362790597435373600253241, −7.87008316371232219607943854656, −7.01672139286090028010307166788, −6.43932138328005097228300610203, −5.10370859604821759964152603807, −4.84636320620368583641397136746, −3.35319196574850752564523700729, −2.75928420513338035449676370861, −2.02260825595865357176947328691,
0.960144696500970110215073422554, 2.14683871022125659305855101612, 3.59480568209050100968620135652, 4.47665030007077433981761781770, 5.35043319747747978150538392354, 5.93848404590964669414696044917, 6.51825347041657159913887740277, 7.63597556496848261987079259800, 8.309486094269842298463383528655, 9.434361504450751619904901618941
