L(s) = 1 | − 2.23·2-s + 3.00·4-s − 2·5-s − 2.23·8-s + 4.47·10-s + 11-s − 3.23·13-s − 0.999·16-s − 3.23·17-s − 6.47·19-s − 6.00·20-s − 2.23·22-s − 2.47·23-s − 25-s + 7.23·26-s − 8.47·29-s + 2.76·31-s + 6.70·32-s + 7.23·34-s − 8.47·37-s + 14.4·38-s + 4.47·40-s − 11.2·41-s + 8·43-s + 3.00·44-s + 5.52·46-s + 2.76·47-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 1.50·4-s − 0.894·5-s − 0.790·8-s + 1.41·10-s + 0.301·11-s − 0.897·13-s − 0.249·16-s − 0.784·17-s − 1.48·19-s − 1.34·20-s − 0.476·22-s − 0.515·23-s − 0.200·25-s + 1.41·26-s − 1.57·29-s + 0.496·31-s + 1.18·32-s + 1.24·34-s − 1.39·37-s + 2.34·38-s + 0.707·40-s − 1.75·41-s + 1.21·43-s + 0.452·44-s + 0.815·46-s + 0.403·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1869874643\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1869874643\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 2.76T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 0.763T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.390489900448908492077675271510, −7.74635118263087205985051906570, −7.00917811335554784659733751534, −6.62362589032368247424947313357, −5.45254262090484139526955638860, −4.38407573056106415985996310991, −3.77464846840460116956361360403, −2.42545747434980519135615613371, −1.76866490657923909616676046718, −0.29032680343432583193398379652,
0.29032680343432583193398379652, 1.76866490657923909616676046718, 2.42545747434980519135615613371, 3.77464846840460116956361360403, 4.38407573056106415985996310991, 5.45254262090484139526955638860, 6.62362589032368247424947313357, 7.00917811335554784659733751534, 7.74635118263087205985051906570, 8.390489900448908492077675271510