L(s) = 1 | − 2-s + 4-s + 3.27·5-s + 4.39·7-s − 8-s − 3.27·10-s + 2.69·11-s + 3.54·13-s − 4.39·14-s + 16-s + 2.86·17-s − 3.06·19-s + 3.27·20-s − 2.69·22-s − 4.38·23-s + 5.73·25-s − 3.54·26-s + 4.39·28-s + 10.5·29-s − 9.78·31-s − 32-s − 2.86·34-s + 14.4·35-s + 2.90·37-s + 3.06·38-s − 3.27·40-s − 1.53·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.46·5-s + 1.66·7-s − 0.353·8-s − 1.03·10-s + 0.812·11-s + 0.984·13-s − 1.17·14-s + 0.250·16-s + 0.695·17-s − 0.703·19-s + 0.732·20-s − 0.574·22-s − 0.915·23-s + 1.14·25-s − 0.695·26-s + 0.831·28-s + 1.95·29-s − 1.75·31-s − 0.176·32-s − 0.491·34-s + 2.43·35-s + 0.477·37-s + 0.497·38-s − 0.517·40-s − 0.239·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.987393433\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.987393433\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 - 2.86T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 9.78T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 + 1.53T + 41T^{2} \) |
| 43 | \( 1 + 9.57T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 - 2.73T + 53T^{2} \) |
| 59 | \( 1 - 8.65T + 59T^{2} \) |
| 61 | \( 1 - 0.791T + 61T^{2} \) |
| 67 | \( 1 + 4.78T + 67T^{2} \) |
| 71 | \( 1 + 9.73T + 71T^{2} \) |
| 73 | \( 1 - 9.30T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 5.84T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.190181581101930706034701450075, −7.13894851460457497561384255905, −6.41274641128732869326137886910, −5.87048770742096248777177089602, −5.19856920685681456914454603949, −4.38439868169922277634340456153, −3.40593786930872012621328784880, −2.17904950367073906202885833186, −1.69228550498617381094804868879, −1.07038863814955948904311036160,
1.07038863814955948904311036160, 1.69228550498617381094804868879, 2.17904950367073906202885833186, 3.40593786930872012621328784880, 4.38439868169922277634340456153, 5.19856920685681456914454603949, 5.87048770742096248777177089602, 6.41274641128732869326137886910, 7.13894851460457497561384255905, 8.190181581101930706034701450075