| L(s) = 1 | + 0.770·3-s − 3.70·5-s + 0.0752·7-s − 2.40·9-s + 0.504·11-s − 1.94·13-s − 2.85·15-s + 0.0239·17-s + 5.16·19-s + 0.0579·21-s − 3.20·23-s + 8.71·25-s − 4.16·27-s − 7.56·29-s − 3.48·31-s + 0.388·33-s − 0.278·35-s − 2.28·37-s − 1.50·39-s − 9.70·43-s + 8.91·45-s + 11.1·47-s − 6.99·49-s + 0.0184·51-s + 10.4·53-s − 1.86·55-s + 3.97·57-s + ⋯ |
| L(s) = 1 | + 0.444·3-s − 1.65·5-s + 0.0284·7-s − 0.802·9-s + 0.152·11-s − 0.540·13-s − 0.736·15-s + 0.00582·17-s + 1.18·19-s + 0.0126·21-s − 0.667·23-s + 1.74·25-s − 0.801·27-s − 1.40·29-s − 0.625·31-s + 0.0676·33-s − 0.0471·35-s − 0.375·37-s − 0.240·39-s − 1.47·43-s + 1.32·45-s + 1.62·47-s − 0.999·49-s + 0.00258·51-s + 1.43·53-s − 0.252·55-s + 0.526·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8671629001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8671629001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 - 0.770T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 - 0.0752T + 7T^{2} \) |
| 11 | \( 1 - 0.504T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 - 0.0239T + 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 23 | \( 1 + 3.20T + 23T^{2} \) |
| 29 | \( 1 + 7.56T + 29T^{2} \) |
| 31 | \( 1 + 3.48T + 31T^{2} \) |
| 37 | \( 1 + 2.28T + 37T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 9.39T + 59T^{2} \) |
| 61 | \( 1 + 8.69T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 + 8.20T + 73T^{2} \) |
| 79 | \( 1 + 7.71T + 79T^{2} \) |
| 83 | \( 1 - 8.88T + 83T^{2} \) |
| 89 | \( 1 - 0.280T + 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
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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
−7.82799049841156536147301830442, −7.54399229319034659538245237506, −6.84753327973124292619996215727, −5.74950722789600714290043155707, −5.10986968673504361545560102781, −4.17785385490762094703156605984, −3.53694676151248457549839022187, −3.01504891631314508036107537017, −1.89687386031743935325413432857, −0.44892007654201318976522735472,
0.44892007654201318976522735472, 1.89687386031743935325413432857, 3.01504891631314508036107537017, 3.53694676151248457549839022187, 4.17785385490762094703156605984, 5.10986968673504361545560102781, 5.74950722789600714290043155707, 6.84753327973124292619996215727, 7.54399229319034659538245237506, 7.82799049841156536147301830442
