| L(s) = 1 | − 2-s − 0.0893·3-s + 4-s + 3.51·5-s + 0.0893·6-s + 1.65·7-s − 8-s − 2.99·9-s − 3.51·10-s + 1.78·11-s − 0.0893·12-s + 3.94·13-s − 1.65·14-s − 0.314·15-s + 16-s + 3.37·17-s + 2.99·18-s + 5.35·19-s + 3.51·20-s − 0.147·21-s − 1.78·22-s − 1.34·23-s + 0.0893·24-s + 7.37·25-s − 3.94·26-s + 0.535·27-s + 1.65·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0515·3-s + 0.5·4-s + 1.57·5-s + 0.0364·6-s + 0.623·7-s − 0.353·8-s − 0.997·9-s − 1.11·10-s + 0.537·11-s − 0.0257·12-s + 1.09·13-s − 0.441·14-s − 0.0811·15-s + 0.250·16-s + 0.818·17-s + 0.705·18-s + 1.22·19-s + 0.786·20-s − 0.0321·21-s − 0.379·22-s − 0.280·23-s + 0.0182·24-s + 1.47·25-s − 0.773·26-s + 0.103·27-s + 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.243662206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.243662206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0893T + 3T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 - 3.37T + 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 + 1.34T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 - 6.04T + 31T^{2} \) |
| 37 | \( 1 + 7.28T + 37T^{2} \) |
| 41 | \( 1 + 2.71T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 2.29T + 47T^{2} \) |
| 53 | \( 1 + 5.40T + 53T^{2} \) |
| 59 | \( 1 - 0.104T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 8.88T + 67T^{2} \) |
| 71 | \( 1 + 3.47T + 71T^{2} \) |
| 73 | \( 1 - 4.52T + 73T^{2} \) |
| 79 | \( 1 - 0.220T + 79T^{2} \) |
| 83 | \( 1 - 6.98T + 83T^{2} \) |
| 89 | \( 1 + 6.96T + 89T^{2} \) |
| 97 | \( 1 - 4.13T + 97T^{2} \) |
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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
−8.415070948430553328209677260522, −8.074410829472379326553152118928, −6.78008691948488073163730751618, −6.33689586258778539955307772370, −5.50523476732420968587574786530, −5.10260622923405655474680649763, −3.56058439273088645839644149987, −2.76219404171793706122072587387, −1.70441838507283781052404560139, −1.06332280518426690504603293818,
1.06332280518426690504603293818, 1.70441838507283781052404560139, 2.76219404171793706122072587387, 3.56058439273088645839644149987, 5.10260622923405655474680649763, 5.50523476732420968587574786530, 6.33689586258778539955307772370, 6.78008691948488073163730751618, 8.074410829472379326553152118928, 8.415070948430553328209677260522
