| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 2·7-s − 8-s + 9-s + 12-s + 2·14-s + 16-s + 17-s − 18-s − 2·21-s + 4·23-s − 24-s + 27-s − 2·28-s − 2·29-s + 4·31-s − 32-s − 34-s + 36-s + 2·37-s + 2·42-s − 6·43-s − 4·46-s + 48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.436·21-s + 0.834·23-s − 0.204·24-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s + 0.328·37-s + 0.308·42-s − 0.914·43-s − 0.589·46-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.529365635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.529365635\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.66879015500768, −12.21976467981860, −11.65544821771447, −11.29875181227812, −10.71145512497535, −10.18977845581544, −9.885694140197355, −9.460989753231248, −9.008667666929270, −8.501402150591875, −8.193092110924614, −7.578278331221358, −7.136129942679209, −6.666875545265024, −6.338909535137428, −5.571503127485891, −5.211871861055249, −4.448848519524448, −3.859931381783970, −3.407267496238186, −2.703824621269193, −2.561911448870822, −1.621134002188931, −1.151436167821269, −0.3715131147786168,
0.3715131147786168, 1.151436167821269, 1.621134002188931, 2.561911448870822, 2.703824621269193, 3.407267496238186, 3.859931381783970, 4.448848519524448, 5.211871861055249, 5.571503127485891, 6.338909535137428, 6.666875545265024, 7.136129942679209, 7.578278331221358, 8.193092110924614, 8.501402150591875, 9.008667666929270, 9.460989753231248, 9.885694140197355, 10.18977845581544, 10.71145512497535, 11.29875181227812, 11.65544821771447, 12.21976467981860, 12.66879015500768
