L(s) = 1 | + 2.53·2-s − 3-s + 4.43·4-s − 2.52·5-s − 2.53·6-s + 6.17·8-s + 9-s − 6.39·10-s − 5.97·11-s − 4.43·12-s − 13-s + 2.52·15-s + 6.80·16-s − 2.79·17-s + 2.53·18-s − 2.01·19-s − 11.1·20-s − 15.1·22-s − 7.07·23-s − 6.17·24-s + 1.36·25-s − 2.53·26-s − 27-s + 4.59·29-s + 6.39·30-s + 2.36·31-s + 4.89·32-s + ⋯ |
L(s) = 1 | + 1.79·2-s − 0.577·3-s + 2.21·4-s − 1.12·5-s − 1.03·6-s + 2.18·8-s + 0.333·9-s − 2.02·10-s − 1.80·11-s − 1.28·12-s − 0.277·13-s + 0.651·15-s + 1.70·16-s − 0.677·17-s + 0.597·18-s − 0.462·19-s − 2.50·20-s − 3.23·22-s − 1.47·23-s − 1.26·24-s + 0.272·25-s − 0.497·26-s − 0.192·27-s + 0.853·29-s + 1.16·30-s + 0.425·31-s + 0.865·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 11 | \( 1 + 5.97T + 11T^{2} \) |
| 17 | \( 1 + 2.79T + 17T^{2} \) |
| 19 | \( 1 + 2.01T + 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 - 4.59T + 29T^{2} \) |
| 31 | \( 1 - 2.36T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 + 8.19T + 41T^{2} \) |
| 43 | \( 1 - 3.59T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 4.79T + 61T^{2} \) |
| 67 | \( 1 - 5.31T + 67T^{2} \) |
| 71 | \( 1 + 8.35T + 71T^{2} \) |
| 73 | \( 1 + 3.32T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 3.68T + 89T^{2} \) |
| 97 | \( 1 - 1.29T + 97T^{2} \) |
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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
−8.421334079971282306601989967042, −7.72679166180468675180869573123, −7.04934966970628173204739160057, −6.13753907503378119565291015908, −5.44469369437926485382464611461, −4.57137890012902838309875047146, −4.16868396679916656940704289331, −3.05027194849717414679674843998, −2.20171947842518313308468754977, 0,
2.20171947842518313308468754977, 3.05027194849717414679674843998, 4.16868396679916656940704289331, 4.57137890012902838309875047146, 5.44469369437926485382464611461, 6.13753907503378119565291015908, 7.04934966970628173204739160057, 7.72679166180468675180869573123, 8.421334079971282306601989967042