| L(s) = 1 | − 2-s + 4-s + 5-s + 5·7-s − 8-s − 10-s − 13-s − 5·14-s + 16-s + 6·17-s + 5·19-s + 20-s − 9·23-s + 25-s + 26-s + 5·28-s − 4·31-s − 32-s − 6·34-s + 5·35-s − 10·37-s − 5·38-s − 40-s − 3·41-s + 8·43-s + 9·46-s − 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.88·7-s − 0.353·8-s − 0.316·10-s − 0.277·13-s − 1.33·14-s + 1/4·16-s + 1.45·17-s + 1.14·19-s + 0.223·20-s − 1.87·23-s + 1/5·25-s + 0.196·26-s + 0.944·28-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.845·35-s − 1.64·37-s − 0.811·38-s − 0.158·40-s − 0.468·41-s + 1.21·43-s + 1.32·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.524023253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.524023253\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−10.20295832867486108634349031646, −9.473812259560348980429745442124, −8.388162226041374703892252125543, −7.86566828039851573836004453407, −7.12559786726867706746830133866, −5.66421127299809088989589629204, −5.17636996627577623726694783444, −3.76718352791364041563854587690, −2.20029727063878084208215448489, −1.28056241416581044677853894546,
1.28056241416581044677853894546, 2.20029727063878084208215448489, 3.76718352791364041563854587690, 5.17636996627577623726694783444, 5.66421127299809088989589629204, 7.12559786726867706746830133866, 7.86566828039851573836004453407, 8.388162226041374703892252125543, 9.473812259560348980429745442124, 10.20295832867486108634349031646
