| L(s) = 1 | + 3-s + (1.73 + 1.41i)5-s + 9-s + 3.46·13-s + (1.73 + 1.41i)15-s + 4.89i·17-s + 4.89i·19-s − 2.82i·23-s + (0.999 + 4.89i)25-s + 27-s − 2.82i·29-s − 6.92·31-s − 3.46·37-s + 3.46·39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (0.774 + 0.632i)5-s + 0.333·9-s + 0.960·13-s + (0.447 + 0.365i)15-s + 1.18i·17-s + 1.12i·19-s − 0.589i·23-s + (0.199 + 0.979i)25-s + 0.192·27-s − 0.525i·29-s − 1.24·31-s − 0.569·37-s + 0.554·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.580346512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.580346512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-1.73 - 1.41i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 9.79iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 9.79iT - 97T^{2} \) |
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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
−9.321898112610400730244374546737, −8.522547502561239657054099929122, −7.86844229624830578425673780624, −6.90351833114970081178327778628, −6.10995253655286131337639419282, −5.54807531185475514159424860584, −4.09963471401582069397778877666, −3.47904060812015272775887617782, −2.33746022963964035057425243479, −1.47878057193348906174539919589,
0.938429746541981741499062042908, 2.08962025653039169046013639394, 3.06879426769570744407815693824, 4.13817831664393473590532665697, 5.09950408030895172221473900892, 5.77525976990973628426618540501, 6.82940786572937409989699413147, 7.51367166177884435848255877464, 8.567868714168875073794809630814, 9.128520846640038780594352508296
