| L(s) = 1 | − 1.52·2-s − 3-s + 0.312·4-s + 4.21·5-s + 1.52·6-s − 4.25·7-s + 2.56·8-s + 9-s − 6.40·10-s − 4.00·11-s − 0.312·12-s + 0.621·13-s + 6.47·14-s − 4.21·15-s − 4.52·16-s − 17-s − 1.52·18-s − 2.65·19-s + 1.31·20-s + 4.25·21-s + 6.09·22-s + 1.86·23-s − 2.56·24-s + 12.7·25-s − 0.944·26-s − 27-s − 1.33·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 0.577·3-s + 0.156·4-s + 1.88·5-s + 0.620·6-s − 1.60·7-s + 0.907·8-s + 0.333·9-s − 2.02·10-s − 1.20·11-s − 0.0903·12-s + 0.172·13-s + 1.73·14-s − 1.08·15-s − 1.13·16-s − 0.242·17-s − 0.358·18-s − 0.608·19-s + 0.294·20-s + 0.929·21-s + 1.29·22-s + 0.388·23-s − 0.523·24-s + 2.54·25-s − 0.185·26-s − 0.192·27-s − 0.251·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 1.52T + 2T^{2} \) |
| 5 | \( 1 - 4.21T + 5T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 11 | \( 1 + 4.00T + 11T^{2} \) |
| 13 | \( 1 - 0.621T + 13T^{2} \) |
| 19 | \( 1 + 2.65T + 19T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 - 0.0363T + 29T^{2} \) |
| 31 | \( 1 - 9.91T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 + 0.793T + 41T^{2} \) |
| 43 | \( 1 + 5.86T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 + 6.68T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 + 5.17T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 3.36T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.309120681925868524737009993857, −7.28131756862927193375209154232, −6.42840343574475639204281129164, −6.17619608789268186164213465319, −5.24952596319880229544639994495, −4.52032657664373199229713647327, −3.00431607132005366050104854652, −2.29760472142614707243931006872, −1.15880679956967069521814043866, 0,
1.15880679956967069521814043866, 2.29760472142614707243931006872, 3.00431607132005366050104854652, 4.52032657664373199229713647327, 5.24952596319880229544639994495, 6.17619608789268186164213465319, 6.42840343574475639204281129164, 7.28131756862927193375209154232, 8.309120681925868524737009993857
