| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 4.70·7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 2.70·13-s − 4.70·14-s − 15-s + 16-s + 17-s + 18-s − 0.701·19-s + 20-s + 4.70·21-s − 22-s − 4.70·23-s − 24-s + 25-s + 2.70·26-s − 27-s − 4.70·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.77·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.749·13-s − 1.25·14-s − 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.160·19-s + 0.223·20-s + 1.02·21-s − 0.213·22-s − 0.980·23-s − 0.204·24-s + 0.200·25-s + 0.529·26-s − 0.192·27-s − 0.888·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.991260278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.991260278\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4.70T + 7T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 19 | \( 1 + 0.701T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 9.40T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 8.10T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 + 1.40T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 8.70T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 4.10T + 97T^{2} \) |
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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
−7.902869649978967835915518669564, −7.17983409218493472169887882074, −6.44346289893681421474503359851, −5.87504534385872603107896358779, −5.62569875596264527145131507213, −4.37671444970624292750185887487, −3.73964035749030762150726116033, −2.96195183225469407336059706662, −2.05617421545381198950561832704, −0.67480439547222293119396811379,
0.67480439547222293119396811379, 2.05617421545381198950561832704, 2.96195183225469407336059706662, 3.73964035749030762150726116033, 4.37671444970624292750185887487, 5.62569875596264527145131507213, 5.87504534385872603107896358779, 6.44346289893681421474503359851, 7.17983409218493472169887882074, 7.902869649978967835915518669564
