| L(s) = 1 | − 3-s − 2.15·5-s − 2.22·7-s + 9-s − 5.26·11-s + 3.01·13-s + 2.15·15-s + 0.872·17-s + 5.13·19-s + 2.22·21-s + 7.30·23-s − 0.371·25-s − 27-s − 6.24·29-s + 10.6·31-s + 5.26·33-s + 4.79·35-s − 5.97·37-s − 3.01·39-s + 5.74·41-s + 8.86·43-s − 2.15·45-s + 6.77·47-s − 2.03·49-s − 0.872·51-s − 10.3·53-s + 11.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.962·5-s − 0.841·7-s + 0.333·9-s − 1.58·11-s + 0.835·13-s + 0.555·15-s + 0.211·17-s + 1.17·19-s + 0.486·21-s + 1.52·23-s − 0.0742·25-s − 0.192·27-s − 1.16·29-s + 1.91·31-s + 0.916·33-s + 0.810·35-s − 0.982·37-s − 0.482·39-s + 0.896·41-s + 1.35·43-s − 0.320·45-s + 0.987·47-s − 0.291·49-s − 0.122·51-s − 1.41·53-s + 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2.15T + 5T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 11 | \( 1 + 5.26T + 11T^{2} \) |
| 13 | \( 1 - 3.01T + 13T^{2} \) |
| 17 | \( 1 - 0.872T + 17T^{2} \) |
| 19 | \( 1 - 5.13T + 19T^{2} \) |
| 23 | \( 1 - 7.30T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 - 5.74T + 41T^{2} \) |
| 43 | \( 1 - 8.86T + 43T^{2} \) |
| 47 | \( 1 - 6.77T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 8.89T + 59T^{2} \) |
| 61 | \( 1 - 4.06T + 61T^{2} \) |
| 67 | \( 1 + 6.51T + 67T^{2} \) |
| 71 | \( 1 + 7.06T + 71T^{2} \) |
| 73 | \( 1 + 4.80T + 73T^{2} \) |
| 79 | \( 1 + 1.01T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−7.69826244575829103478224848625, −7.62169017135899414593970595336, −6.60710744246709708031814761523, −5.80332039627955871574328253677, −5.15639911039439347087149994910, −4.29386614866074812147948451023, −3.33695640703528301999421908205, −2.77027600087297631741941913532, −1.07960617300720810063433122728, 0,
1.07960617300720810063433122728, 2.77027600087297631741941913532, 3.33695640703528301999421908205, 4.29386614866074812147948451023, 5.15639911039439347087149994910, 5.80332039627955871574328253677, 6.60710744246709708031814761523, 7.62169017135899414593970595336, 7.69826244575829103478224848625
