| L(s) = 1 | + 1.96e4·3-s − 7.90e6·5-s − 1.07e8·7-s + 3.87e8·9-s + 1.01e10·11-s + 2.11e10·13-s − 1.55e11·15-s − 1.45e11·17-s + 1.97e12·19-s − 2.11e12·21-s − 7.44e12·23-s + 4.34e13·25-s + 7.62e12·27-s + 1.27e14·29-s − 1.80e14·31-s + 2.00e14·33-s + 8.51e14·35-s + 1.42e15·37-s + 4.16e14·39-s − 7.58e14·41-s + 2.46e15·43-s − 3.06e15·45-s − 1.33e15·47-s + 1.83e14·49-s − 2.85e15·51-s − 1.45e16·53-s − 8.04e16·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.81·5-s − 1.00·7-s + 0.333·9-s + 1.30·11-s + 0.553·13-s − 1.04·15-s − 0.297·17-s + 1.40·19-s − 0.581·21-s − 0.861·23-s + 2.27·25-s + 0.192·27-s + 1.62·29-s − 1.22·31-s + 0.750·33-s + 1.82·35-s + 1.79·37-s + 0.319·39-s − 0.361·41-s + 0.747·43-s − 0.603·45-s − 0.174·47-s + 0.0161·49-s − 0.171·51-s − 0.606·53-s − 2.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(1.566215259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.566215259\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.96e4T \) |
| good | 5 | \( 1 + 7.90e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 1.07e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.01e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 2.11e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 1.45e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.97e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 7.44e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.27e14T + 6.10e27T^{2} \) |
| 31 | \( 1 + 1.80e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.42e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 7.58e14T + 4.39e30T^{2} \) |
| 43 | \( 1 - 2.46e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 1.33e15T + 5.88e31T^{2} \) |
| 53 | \( 1 + 1.45e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 3.40e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 2.57e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 2.42e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 6.23e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 8.87e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.04e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.43e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.94e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 8.65e18T + 5.60e37T^{2} \) |
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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
−15.62164344509080877074104385044, −14.20893154034598802313486928285, −12.50143962273676612315896720316, −11.39414112459301864202272961485, −9.410780910208745748662419990785, −8.039421958348445779008322391608, −6.70705275055423707714254690683, −4.10378332790324489590619799077, −3.24412154110500059541335269942, −0.794838129351415917045789191823,
0.794838129351415917045789191823, 3.24412154110500059541335269942, 4.10378332790324489590619799077, 6.70705275055423707714254690683, 8.039421958348445779008322391608, 9.410780910208745748662419990785, 11.39414112459301864202272961485, 12.50143962273676612315896720316, 14.20893154034598802313486928285, 15.62164344509080877074104385044
