| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 6·11-s − 4·13-s − 2·14-s + 16-s + 6·17-s + 19-s + 20-s + 6·22-s + 25-s + 4·26-s + 2·28-s + 8·31-s − 32-s − 6·34-s + 2·35-s + 8·37-s − 38-s − 40-s + 12·41-s + 2·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 1.80·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.223·20-s + 1.27·22-s + 1/5·25-s + 0.784·26-s + 0.377·28-s + 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.338·35-s + 1.31·37-s − 0.162·38-s − 0.158·40-s + 1.87·41-s + 0.304·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.303198289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.303198289\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.498573532652488771385352472634, −8.396120495186312555468337461732, −7.71815239727995297209637936624, −7.40324461827821970052106158768, −6.02018801955886064007060925313, −5.34546022558899141381965141097, −4.56694930325129271519942535650, −2.91133319867084723778855091440, −2.30283657246697061436212829968, −0.879125982758298393177104864028,
0.879125982758298393177104864028, 2.30283657246697061436212829968, 2.91133319867084723778855091440, 4.56694930325129271519942535650, 5.34546022558899141381965141097, 6.02018801955886064007060925313, 7.40324461827821970052106158768, 7.71815239727995297209637936624, 8.396120495186312555468337461732, 9.498573532652488771385352472634
