| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 1.23·7-s − 8-s + 9-s − 4.47·11-s + 12-s + 13-s + 1.23·14-s + 16-s + 2.76·17-s − 18-s − 7.23·19-s − 1.23·21-s + 4.47·22-s + 7.23·23-s − 24-s − 26-s + 27-s − 1.23·28-s + 9.70·29-s − 4·31-s − 32-s − 4.47·33-s − 2.76·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.467·7-s − 0.353·8-s + 0.333·9-s − 1.34·11-s + 0.288·12-s + 0.277·13-s + 0.330·14-s + 0.250·16-s + 0.670·17-s − 0.235·18-s − 1.66·19-s − 0.269·21-s + 0.953·22-s + 1.50·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.233·28-s + 1.80·29-s − 0.718·31-s − 0.176·32-s − 0.778·33-s − 0.474·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.340378016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.340378016\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 0.472T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−9.175324169071292800701722032760, −8.339988887619988728505937108720, −7.88514352015359789246426403235, −6.97460868679351519641377351425, −6.22418461228716032422625043520, −5.21020202523382852360493226852, −4.13339410802877928041518753162, −2.93866914531849251157928776443, −2.38020811421580125519167545393, −0.829834607169383335180986414194,
0.829834607169383335180986414194, 2.38020811421580125519167545393, 2.93866914531849251157928776443, 4.13339410802877928041518753162, 5.21020202523382852360493226852, 6.22418461228716032422625043520, 6.97460868679351519641377351425, 7.88514352015359789246426403235, 8.339988887619988728505937108720, 9.175324169071292800701722032760
