L(s) = 1 | + 2i·2-s − 4·4-s + 29.5i·7-s − 8i·8-s − 46.5·11-s + 92.0i·13-s − 59.0·14-s + 16·16-s − 4.53i·17-s − 87.5·19-s − 93.0i·22-s − 160. i·23-s − 184.·26-s − 118. i·28-s − 241.·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.59i·7-s − 0.353i·8-s − 1.27·11-s + 1.96i·13-s − 1.12·14-s + 0.250·16-s − 0.0647i·17-s − 1.05·19-s − 0.901i·22-s − 1.45i·23-s − 1.38·26-s − 0.797i·28-s − 1.54·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1953243093\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1953243093\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 29.5iT - 343T^{2} \) |
| 11 | \( 1 + 46.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 92.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.53iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 87.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 160. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.68T + 2.97e4T^{2} \) |
| 37 | \( 1 - 20.6iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 501.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 294. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 478. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 243. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 383.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 132.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 582. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 566.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 839. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 451.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 301. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 739.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.14e3iT - 9.12e5T^{2} \) |
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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
−9.393114681324644429363752571862, −9.147079827447861197473349222281, −8.287051936291418077199311548307, −7.53811911762226991541906710173, −6.38994778471479198009879329820, −5.99085261939686463312898525102, −4.92382291652124172997098713968, −4.26649619733231476727347546058, −2.70064559782080553064572950336, −1.97153335911793548683504546136,
0.05523780219833771767436212974, 0.837466679985845085924635126832, 2.20507243509820127050795488789, 3.35244264243805803148364687405, 3.98359432289464594331800742606, 5.15239751428240209354505118023, 5.79321996217735920997076546791, 7.32393548612553855961698234348, 7.67965959235571825656848035582, 8.512251218651421751541729911104