L(s) = 1 | − 2-s + 4-s − 3.16·5-s − 7-s − 8-s + 3.16·10-s + 11-s + 2·13-s + 14-s + 16-s − 0.837·17-s − 1.16·19-s − 3.16·20-s − 22-s − 6.32·23-s + 5.00·25-s − 2·26-s − 28-s − 4·29-s + 2.83·31-s − 32-s + 0.837·34-s + 3.16·35-s − 4.32·37-s + 1.16·38-s + 3.16·40-s + 0.837·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.41·5-s − 0.377·7-s − 0.353·8-s + 1.00·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 0.250·16-s − 0.203·17-s − 0.266·19-s − 0.707·20-s − 0.213·22-s − 1.31·23-s + 1.00·25-s − 0.392·26-s − 0.188·28-s − 0.742·29-s + 0.509·31-s − 0.176·32-s + 0.143·34-s + 0.534·35-s − 0.710·37-s + 0.188·38-s + 0.500·40-s + 0.130·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7035990175\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7035990175\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 3.16T + 5T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 0.837T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 - 0.837T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 - 7.48T + 47T^{2} \) |
| 53 | \( 1 - 8.32T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 8.32T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 + 1.16T + 83T^{2} \) |
| 89 | \( 1 + 3.67T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−9.479154363203128727447259177332, −8.633398914850368824601514782239, −8.076332270978591892969200948968, −7.29101283497002339709785263193, −6.55564655002524323292668834712, −5.56084083773721755730789320532, −4.10819024174616492266277601488, −3.65288355157941774785309992502, −2.27965959149649790550620084538, −0.65731802928969972534507916828,
0.65731802928969972534507916828, 2.27965959149649790550620084538, 3.65288355157941774785309992502, 4.10819024174616492266277601488, 5.56084083773721755730789320532, 6.55564655002524323292668834712, 7.29101283497002339709785263193, 8.076332270978591892969200948968, 8.633398914850368824601514782239, 9.479154363203128727447259177332