| L(s) = 1 | − 1.60·2-s + 2.52·3-s + 0.577·4-s − 5-s − 4.06·6-s − 3.16·7-s + 2.28·8-s + 3.39·9-s + 1.60·10-s + 11-s + 1.46·12-s + 0.868·13-s + 5.07·14-s − 2.52·15-s − 4.82·16-s − 5.38·17-s − 5.45·18-s + 5.17·19-s − 0.577·20-s − 7.99·21-s − 1.60·22-s − 6.99·23-s + 5.77·24-s + 25-s − 1.39·26-s + 1.00·27-s − 1.82·28-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 1.46·3-s + 0.288·4-s − 0.447·5-s − 1.65·6-s − 1.19·7-s + 0.807·8-s + 1.13·9-s + 0.507·10-s + 0.301·11-s + 0.421·12-s + 0.240·13-s + 1.35·14-s − 0.653·15-s − 1.20·16-s − 1.30·17-s − 1.28·18-s + 1.18·19-s − 0.129·20-s − 1.74·21-s − 0.342·22-s − 1.45·23-s + 1.17·24-s + 0.200·25-s − 0.273·26-s + 0.194·27-s − 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.60T + 2T^{2} \) |
| 3 | \( 1 - 2.52T + 3T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 13 | \( 1 - 0.868T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 + 6.99T + 23T^{2} \) |
| 29 | \( 1 - 9.43T + 29T^{2} \) |
| 31 | \( 1 - 9.37T + 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 + 9.21T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 8.78T + 67T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 6.71T + 83T^{2} \) |
| 89 | \( 1 + 3.73T + 89T^{2} \) |
| 97 | \( 1 - 9.23T + 97T^{2} \) |
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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
−8.295829398651725306437081184728, −7.75643092481844768332570907410, −6.79611223845218190167691115779, −6.38552869375425573921652433478, −4.78421187479973570685356525622, −4.02864460165939643644891517711, −3.21292657790893444047206707402, −2.49156341438408872482076159226, −1.33925080886121949104208474013, 0,
1.33925080886121949104208474013, 2.49156341438408872482076159226, 3.21292657790893444047206707402, 4.02864460165939643644891517711, 4.78421187479973570685356525622, 6.38552869375425573921652433478, 6.79611223845218190167691115779, 7.75643092481844768332570907410, 8.295829398651725306437081184728
