| L(s) = 1 | + 2.76·2-s − 3-s + 5.65·4-s + 0.300·5-s − 2.76·6-s + 10.1·8-s + 9-s + 0.832·10-s + 1.37·11-s − 5.65·12-s + 13-s − 0.300·15-s + 16.7·16-s + 4.90·17-s + 2.76·18-s − 1.43·19-s + 1.70·20-s + 3.79·22-s − 5.39·23-s − 10.1·24-s − 4.90·25-s + 2.76·26-s − 27-s − 6.32·29-s − 0.832·30-s − 6.83·31-s + 25.9·32-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 0.577·3-s + 2.82·4-s + 0.134·5-s − 1.12·6-s + 3.58·8-s + 0.333·9-s + 0.263·10-s + 0.413·11-s − 1.63·12-s + 0.277·13-s − 0.0776·15-s + 4.17·16-s + 1.18·17-s + 0.652·18-s − 0.329·19-s + 0.380·20-s + 0.809·22-s − 1.12·23-s − 2.06·24-s − 0.981·25-s + 0.542·26-s − 0.192·27-s − 1.17·29-s − 0.151·30-s − 1.22·31-s + 4.59·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.699959059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.699959059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 5 | \( 1 - 0.300T + 5T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 17 | \( 1 - 4.90T + 17T^{2} \) |
| 19 | \( 1 + 1.43T + 19T^{2} \) |
| 23 | \( 1 + 5.39T + 23T^{2} \) |
| 29 | \( 1 + 6.32T + 29T^{2} \) |
| 31 | \( 1 + 6.83T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 0.336T + 41T^{2} \) |
| 43 | \( 1 - 1.87T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 3.80T + 53T^{2} \) |
| 59 | \( 1 + 6.17T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 8.26T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 - 7.32T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−9.486618350352775366571415705488, −7.899660406330154966472352406821, −7.37969367496121225904733686330, −6.37289170994933157328339119071, −5.79098544995923325092853018573, −5.34167806659739584994567298057, −4.07918035797317145883193614751, −3.82718424686451471691273661704, −2.51256246517904218872235446988, −1.48538626003433748983521866911,
1.48538626003433748983521866911, 2.51256246517904218872235446988, 3.82718424686451471691273661704, 4.07918035797317145883193614751, 5.34167806659739584994567298057, 5.79098544995923325092853018573, 6.37289170994933157328339119071, 7.37969367496121225904733686330, 7.899660406330154966472352406821, 9.486618350352775366571415705488
