L(s) = 1 | + 2-s − 3-s + 4-s + 2.74·5-s − 6-s − 2.39·7-s + 8-s + 9-s + 2.74·10-s − 1.84·11-s − 12-s + 13-s − 2.39·14-s − 2.74·15-s + 16-s − 7.09·17-s + 18-s + 7.61·19-s + 2.74·20-s + 2.39·21-s − 1.84·22-s − 0.525·23-s − 24-s + 2.56·25-s + 26-s − 27-s − 2.39·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.22·5-s − 0.408·6-s − 0.906·7-s + 0.353·8-s + 0.333·9-s + 0.869·10-s − 0.557·11-s − 0.288·12-s + 0.277·13-s − 0.641·14-s − 0.710·15-s + 0.250·16-s − 1.72·17-s + 0.235·18-s + 1.74·19-s + 0.614·20-s + 0.523·21-s − 0.394·22-s − 0.109·23-s − 0.204·24-s + 0.512·25-s + 0.196·26-s − 0.192·27-s − 0.453·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 2.74T + 5T^{2} \) |
| 7 | \( 1 + 2.39T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 17 | \( 1 + 7.09T + 17T^{2} \) |
| 19 | \( 1 - 7.61T + 19T^{2} \) |
| 23 | \( 1 + 0.525T + 23T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 + 4.81T + 31T^{2} \) |
| 37 | \( 1 + 3.48T + 37T^{2} \) |
| 41 | \( 1 - 0.958T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.33T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 1.08T + 59T^{2} \) |
| 61 | \( 1 + 8.03T + 61T^{2} \) |
| 67 | \( 1 - 8.95T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + 2.09T + 73T^{2} \) |
| 79 | \( 1 - 2.18T + 79T^{2} \) |
| 83 | \( 1 - 6.22T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + 1.02T + 97T^{2} \) |
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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
−7.05210619178666191731248773823, −6.70348621575413086156633771688, −5.95948984843784912992014104638, −5.45114300861803582170290784696, −4.93364274951969505381923759541, −3.90413672968928116359377201371, −3.12681005487854165774623078699, −2.28334478940358831545298111656, −1.48286346400189866853131050179, 0,
1.48286346400189866853131050179, 2.28334478940358831545298111656, 3.12681005487854165774623078699, 3.90413672968928116359377201371, 4.93364274951969505381923759541, 5.45114300861803582170290784696, 5.95948984843784912992014104638, 6.70348621575413086156633771688, 7.05210619178666191731248773823