L(s) = 1 | + 2-s + 4-s − 5-s + 3·7-s + 8-s − 10-s + 11-s + 6.81·13-s + 3·14-s + 16-s − 17-s − 2.81·19-s − 20-s + 22-s + 4.81·23-s + 25-s + 6.81·26-s + 3·28-s − 3.81·29-s − 3.81·31-s + 32-s − 34-s − 3·35-s + 37-s − 2.81·38-s − 40-s − 5.81·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 1.89·13-s + 0.801·14-s + 0.250·16-s − 0.242·17-s − 0.645·19-s − 0.223·20-s + 0.213·22-s + 1.00·23-s + 0.200·25-s + 1.33·26-s + 0.566·28-s − 0.708·29-s − 0.685·31-s + 0.176·32-s − 0.171·34-s − 0.507·35-s + 0.164·37-s − 0.456·38-s − 0.158·40-s − 0.908·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.552762917\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.552762917\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 6.81T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 2.81T + 19T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 9.63T + 71T^{2} \) |
| 73 | \( 1 - 4.81T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 6.81T + 83T^{2} \) |
| 89 | \( 1 + 1.18T + 89T^{2} \) |
| 97 | \( 1 - 5.81T + 97T^{2} \) |
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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
−8.746234228831373618614876733506, −7.78139614306475178989368712750, −7.14746409340123750275284378144, −6.22681062755661031280744619480, −5.58885133412496466278343278413, −4.65770788562314959731892215257, −4.01913183700136757998998670756, −3.29992962881792263266317472408, −2.02545534892462912699227082759, −1.11225397641977400594990303362,
1.11225397641977400594990303362, 2.02545534892462912699227082759, 3.29992962881792263266317472408, 4.01913183700136757998998670756, 4.65770788562314959731892215257, 5.58885133412496466278343278413, 6.22681062755661031280744619480, 7.14746409340123750275284378144, 7.78139614306475178989368712750, 8.746234228831373618614876733506