| L(s) = 1 | − 2.56·3-s − 1.56·5-s + 3·7-s + 3.56·9-s + 3.56·11-s − 2.56·13-s + 4·15-s − 8.12·17-s − 19-s − 7.68·21-s + 1.43·23-s − 2.56·25-s − 1.43·27-s + 7.68·29-s + 0.876·31-s − 9.12·33-s − 4.68·35-s + 1.12·37-s + 6.56·39-s + 4·41-s − 9.56·43-s − 5.56·45-s + 8.68·47-s + 2·49-s + 20.8·51-s + 8.56·53-s − 5.56·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.698·5-s + 1.13·7-s + 1.18·9-s + 1.07·11-s − 0.710·13-s + 1.03·15-s − 1.97·17-s − 0.229·19-s − 1.67·21-s + 0.299·23-s − 0.512·25-s − 0.276·27-s + 1.42·29-s + 0.157·31-s − 1.58·33-s − 0.791·35-s + 0.184·37-s + 1.05·39-s + 0.624·41-s − 1.45·43-s − 0.829·45-s + 1.26·47-s + 0.285·49-s + 2.91·51-s + 1.17·53-s − 0.749·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8439132088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8439132088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 + 8.12T + 17T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 - 0.876T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 9.56T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 - 8.56T + 59T^{2} \) |
| 61 | \( 1 - 5.80T + 61T^{2} \) |
| 67 | \( 1 - 4.56T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 7.24T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 7.36T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−9.921425562901589940143362404457, −8.818487832315039984289995285059, −8.103142064317069914001051227230, −6.92550031419923925262283436369, −6.57252096985603075527736830354, −5.38001367173071177485282582669, −4.60916292509951064396535128823, −4.07718997671769238739851566286, −2.17863654002928259345487141033, −0.73962667314684707641375373116,
0.73962667314684707641375373116, 2.17863654002928259345487141033, 4.07718997671769238739851566286, 4.60916292509951064396535128823, 5.38001367173071177485282582669, 6.57252096985603075527736830354, 6.92550031419923925262283436369, 8.103142064317069914001051227230, 8.818487832315039984289995285059, 9.921425562901589940143362404457
