L(s) = 1 | − 3-s − 7-s − 2·9-s + 3·11-s + 13-s + 3·17-s + 2·19-s + 21-s + 6·23-s + 5·27-s − 9·29-s + 8·31-s − 3·33-s + 10·37-s − 39-s − 2·43-s + 3·47-s + 49-s − 3·51-s − 2·57-s + 12·59-s + 8·61-s + 2·63-s − 8·67-s − 6·69-s − 14·73-s − 3·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.277·13-s + 0.727·17-s + 0.458·19-s + 0.218·21-s + 1.25·23-s + 0.962·27-s − 1.67·29-s + 1.43·31-s − 0.522·33-s + 1.64·37-s − 0.160·39-s − 0.304·43-s + 0.437·47-s + 1/7·49-s − 0.420·51-s − 0.264·57-s + 1.56·59-s + 1.02·61-s + 0.251·63-s − 0.977·67-s − 0.722·69-s − 1.63·73-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.202379852\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.202379852\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−10.52275298271203907560462125666, −9.551416552558793868042741249641, −8.871423639951469868858510987036, −7.80402753154155331215140932996, −6.76658613317771706160984607640, −5.98473030303912907687600422758, −5.17058967893037473212097767412, −3.90226018894727439504301993022, −2.82281852164300603542955300159, −0.992373800423296370916788271681,
0.992373800423296370916788271681, 2.82281852164300603542955300159, 3.90226018894727439504301993022, 5.17058967893037473212097767412, 5.98473030303912907687600422758, 6.76658613317771706160984607640, 7.80402753154155331215140932996, 8.871423639951469868858510987036, 9.551416552558793868042741249641, 10.52275298271203907560462125666