| L(s) = 1 | − 4.75·2-s − 3·3-s + 14.6·4-s + 14.2·6-s + 31.5·7-s − 31.6·8-s + 9·9-s − 7.18·11-s − 43.9·12-s − 84.3·13-s − 150.·14-s + 33.5·16-s − 17·17-s − 42.8·18-s − 37.0·19-s − 94.7·21-s + 34.2·22-s − 150.·23-s + 95.0·24-s + 401.·26-s − 27·27-s + 462.·28-s − 11.5·29-s − 53.2·31-s + 93.7·32-s + 21.5·33-s + 80.9·34-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.577·3-s + 1.83·4-s + 0.971·6-s + 1.70·7-s − 1.40·8-s + 0.333·9-s − 0.197·11-s − 1.05·12-s − 1.79·13-s − 2.87·14-s + 0.524·16-s − 0.242·17-s − 0.560·18-s − 0.447·19-s − 0.984·21-s + 0.331·22-s − 1.36·23-s + 0.808·24-s + 3.02·26-s − 0.192·27-s + 3.12·28-s − 0.0741·29-s − 0.308·31-s + 0.517·32-s + 0.113·33-s + 0.408·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5595926989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5595926989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 4.75T + 8T^{2} \) |
| 7 | \( 1 - 31.5T + 343T^{2} \) |
| 11 | \( 1 + 7.18T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 53.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 99.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 456.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 571.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 48.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 59.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 740.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 930.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 697.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 22.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 369.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
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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
−9.409671622836960609994421720778, −8.366876102083584655460175745652, −7.78749645300189867005575731867, −7.30439088755871652462056976366, −6.22761468168631575823606606335, −5.09718322736222845322100665268, −4.38908272650641289135800076463, −2.35848337848680608601287708906, −1.75243727053052215101994146731, −0.49919182973047221297047497649,
0.49919182973047221297047497649, 1.75243727053052215101994146731, 2.35848337848680608601287708906, 4.38908272650641289135800076463, 5.09718322736222845322100665268, 6.22761468168631575823606606335, 7.30439088755871652462056976366, 7.78749645300189867005575731867, 8.366876102083584655460175745652, 9.409671622836960609994421720778
