L(s) = 1 | + (0.900 − 1.56i)2-s + (0.623 + 1.07i)3-s + (−1.12 − 1.94i)4-s + 2.24·6-s − 2.24·8-s + (−0.277 + 0.480i)9-s + (1.40 − 2.42i)12-s + 1.80·13-s + (−0.900 + 1.56i)16-s + (−0.222 − 0.385i)17-s + (0.5 + 0.866i)18-s + (−0.222 + 0.385i)19-s + (−0.623 + 1.07i)23-s + (−1.40 − 2.42i)24-s + (−0.5 − 0.866i)25-s + (1.62 − 2.81i)26-s + ⋯ |
L(s) = 1 | + (0.900 − 1.56i)2-s + (0.623 + 1.07i)3-s + (−1.12 − 1.94i)4-s + 2.24·6-s − 2.24·8-s + (−0.277 + 0.480i)9-s + (1.40 − 2.42i)12-s + 1.80·13-s + (−0.900 + 1.56i)16-s + (−0.222 − 0.385i)17-s + (0.5 + 0.866i)18-s + (−0.222 + 0.385i)19-s + (−0.623 + 1.07i)23-s + (−1.40 − 2.42i)24-s + (−0.5 − 0.866i)25-s + (1.62 − 2.81i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.189096510\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.189096510\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.80T + T^{2} \) |
| 17 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.24T + T^{2} \) |
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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
−9.501699021753811911276129019053, −8.794163293411639487801782865412, −7.959681809300946824120904898981, −6.35226389030292845089428189104, −5.62379910538386516296562473044, −4.63375378762089689495879156420, −3.87345243392989194229338668329, −3.54652072770348459017211618959, −2.53557174314562538911207541897, −1.39791643264655361206048294006,
1.65919010299767152506857707676, 3.07571694836446014133026789368, 3.90542753497502932814445841013, 4.82836953762482163783993782437, 5.88521498722584552709073820709, 6.49878255232474630489208330425, 6.95447597875377976741427818740, 7.924379939417520550759319958347, 8.416627160696390653759239728244, 8.775038965134369957015150194482