L(s) = 1 | + 9.35·5-s − 21.2i·7-s + 50.9i·11-s − 33.2i·13-s + 120. i·17-s − 82.1·19-s + 95.3·23-s − 37.4·25-s + 83.1·29-s − 36.4i·31-s − 198. i·35-s − 201. i·37-s + 291. i·41-s − 457.·43-s − 628.·47-s + ⋯ |
L(s) = 1 | + 0.837·5-s − 1.14i·7-s + 1.39i·11-s − 0.709i·13-s + 1.72i·17-s − 0.991·19-s + 0.864·23-s − 0.299·25-s + 0.532·29-s − 0.210i·31-s − 0.958i·35-s − 0.896i·37-s + 1.11i·41-s − 1.62·43-s − 1.95·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8368785339\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8368785339\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 9.35T + 125T^{2} \) |
| 7 | \( 1 + 21.2iT - 343T^{2} \) |
| 11 | \( 1 - 50.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 33.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 120. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 82.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 95.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 83.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 36.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 201. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 291. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 457.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 628.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 659.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 44.8iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 149. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 134.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 291.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 888.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 714. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 827. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 19.6iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 805.T + 9.12e5T^{2} \) |
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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
−9.480421626408609912768059672727, −8.330301521445360024960843866012, −7.73681884756762510941559354667, −6.71106458459353701902541232940, −6.25523155944935097426677061671, −5.07605426115823634650455908569, −4.34835282577419325527387606131, −3.41121994027112339912629064660, −2.07521186517975615602216602730, −1.34490550852125030756225633462,
0.16277547213795581016096174112, 1.58854002979042939330576217709, 2.58206503618285120451842799106, 3.30927200281234981102744210465, 4.81467328826673257685767153455, 5.37734770987870306124416533159, 6.30554622499860108585688270487, 6.76362938029008577356968178502, 8.126229176243342587979264722711, 8.768730263589721046884388713002