| L(s) = 1 | − 4.28·2-s + 3·3-s + 10.3·4-s + 5·5-s − 12.8·6-s + 9.81·7-s − 9.93·8-s + 9·9-s − 21.4·10-s + 30.9·12-s + 85.8·13-s − 42.0·14-s + 15·15-s − 40.0·16-s + 35.0·17-s − 38.5·18-s + 22.0·19-s + 51.6·20-s + 29.4·21-s + 144.·23-s − 29.8·24-s + 25·25-s − 367.·26-s + 27·27-s + 101.·28-s + 195.·29-s − 64.2·30-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 0.577·3-s + 1.29·4-s + 0.447·5-s − 0.873·6-s + 0.529·7-s − 0.439·8-s + 0.333·9-s − 0.676·10-s + 0.744·12-s + 1.83·13-s − 0.801·14-s + 0.258·15-s − 0.625·16-s + 0.500·17-s − 0.504·18-s + 0.265·19-s + 0.576·20-s + 0.305·21-s + 1.30·23-s − 0.253·24-s + 0.200·25-s − 2.77·26-s + 0.192·27-s + 0.683·28-s + 1.25·29-s − 0.390·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.979278273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979278273\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.28T + 8T^{2} \) |
| 7 | \( 1 - 9.81T + 343T^{2} \) |
| 13 | \( 1 - 85.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 22.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 144.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 248.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 297.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 3.87T + 6.89e4T^{2} \) |
| 43 | \( 1 - 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 351.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 219.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 749.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 497.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 628.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 231.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 917.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 528.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 46.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 945.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 903.T + 9.12e5T^{2} \) |
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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
−8.871430505629683827800063831214, −8.332943184499890120558504593124, −7.66455700346647034365944462918, −6.82617664642914021334134406763, −5.97328233800500764588286793100, −4.84175783548526828978280006344, −3.66098998692210296964960835231, −2.58481612144832543169707193857, −1.45476248123963236471206497289, −0.936968631093991228789481630905,
0.936968631093991228789481630905, 1.45476248123963236471206497289, 2.58481612144832543169707193857, 3.66098998692210296964960835231, 4.84175783548526828978280006344, 5.97328233800500764588286793100, 6.82617664642914021334134406763, 7.66455700346647034365944462918, 8.332943184499890120558504593124, 8.871430505629683827800063831214
