| L(s) = 1 | − 2.56i·2-s − 4.56·4-s + 0.438i·7-s + 6.56i·8-s − 1.56·11-s − i·13-s + 1.12·14-s + 7.68·16-s − 1.56i·17-s + 5.12·19-s + 4i·22-s + 2.43i·23-s − 2.56·26-s − 2i·28-s + 7.12·29-s + ⋯ |
| L(s) = 1 | − 1.81i·2-s − 2.28·4-s + 0.165i·7-s + 2.31i·8-s − 0.470·11-s − 0.277i·13-s + 0.300·14-s + 1.92·16-s − 0.378i·17-s + 1.17·19-s + 0.852i·22-s + 0.508i·23-s − 0.502·26-s − 0.377i·28-s + 1.32·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.446602269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.446602269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 + 2.56iT - 2T^{2} \) |
| 7 | \( 1 - 0.438iT - 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 17 | \( 1 + 1.56iT - 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 2.43iT - 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 3.12iT - 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 4.68iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 - 11.3iT - 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 4.68T + 79T^{2} \) |
| 83 | \( 1 + 16.4iT - 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−8.636555408957354431999488313242, −8.059134490060930923849092268214, −7.03816708219163981007124875856, −5.83134494950060801168964073531, −4.96770664861745775233450572262, −4.38556998991923175014353566161, −3.12827078831167771823960325303, −2.89237118834506148333026905213, −1.66074262494975236278290587920, −0.66462675180913818179018144230,
0.874020332012693529786356045394, 2.69569270250612422945595442605, 3.95595790217648379341153352371, 4.68054411481477592986656529344, 5.41345288048321806826107657821, 6.18025790019049081302577031680, 6.75631365046335550018211651444, 7.71672022628864312031082754476, 7.901342193312102699757567088733, 8.939504912440241724357290625193
