| L(s) = 1 | + 0.984·2-s + 3.01·3-s − 1.03·4-s + 5-s + 2.96·6-s + 3.40·7-s − 2.98·8-s + 6.09·9-s + 0.984·10-s + 11-s − 3.10·12-s + 3.41·13-s + 3.35·14-s + 3.01·15-s − 0.877·16-s − 1.53·17-s + 5.99·18-s − 4.91·19-s − 1.03·20-s + 10.2·21-s + 0.984·22-s + 1.63·23-s − 8.99·24-s + 25-s + 3.36·26-s + 9.31·27-s − 3.50·28-s + ⋯ |
| L(s) = 1 | + 0.696·2-s + 1.74·3-s − 0.515·4-s + 0.447·5-s + 1.21·6-s + 1.28·7-s − 1.05·8-s + 2.03·9-s + 0.311·10-s + 0.301·11-s − 0.896·12-s + 0.947·13-s + 0.896·14-s + 0.778·15-s − 0.219·16-s − 0.371·17-s + 1.41·18-s − 1.12·19-s − 0.230·20-s + 2.24·21-s + 0.209·22-s + 0.341·23-s − 1.83·24-s + 0.200·25-s + 0.659·26-s + 1.79·27-s − 0.663·28-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
| \(L(1)\) |
\(\approx\) |
\(5.726645185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.726645185\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.984T + 2T^{2} \) |
| 3 | \( 1 - 3.01T + 3T^{2} \) |
| 7 | \( 1 - 3.40T + 7T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 - 0.688T + 29T^{2} \) |
| 31 | \( 1 - 0.0844T + 31T^{2} \) |
| 37 | \( 1 + 5.26T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 0.878T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 79 | \( 1 - 1.01T + 79T^{2} \) |
| 83 | \( 1 + 4.02T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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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
−8.509514573312491416112088413056, −8.038188804312560617838348419944, −7.07290231324557286146396610079, −6.16659762718305680499077998125, −5.24901355040005012536519390577, −4.32814247546025511910544021883, −4.00611671541923771615333696804, −3.02232919073635727724599437559, −2.18639895320115787963870745776, −1.31792833822744280889477911565,
1.31792833822744280889477911565, 2.18639895320115787963870745776, 3.02232919073635727724599437559, 4.00611671541923771615333696804, 4.32814247546025511910544021883, 5.24901355040005012536519390577, 6.16659762718305680499077998125, 7.07290231324557286146396610079, 8.038188804312560617838348419944, 8.509514573312491416112088413056
