| L(s) = 1 | + 3-s − 2.71·5-s + 0.735·7-s + 9-s + 1.14·11-s − 2.71·15-s + 5.96·17-s − 3.07·19-s + 0.735·21-s + 5.31·23-s + 2.39·25-s + 27-s − 8.12·29-s − 1.67·31-s + 1.14·33-s − 2.00·35-s − 1.90·37-s + 8.69·41-s + 11.7·43-s − 2.71·45-s − 7.53·47-s − 6.45·49-s + 5.96·51-s + 1.86·53-s − 3.10·55-s − 3.07·57-s − 4.28·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.21·5-s + 0.278·7-s + 0.333·9-s + 0.344·11-s − 0.702·15-s + 1.44·17-s − 0.705·19-s + 0.160·21-s + 1.10·23-s + 0.478·25-s + 0.192·27-s − 1.50·29-s − 0.299·31-s + 0.199·33-s − 0.338·35-s − 0.313·37-s + 1.35·41-s + 1.78·43-s − 0.405·45-s − 1.09·47-s − 0.922·49-s + 0.835·51-s + 0.255·53-s − 0.419·55-s − 0.407·57-s − 0.557·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.956773646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.956773646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.71T + 5T^{2} \) |
| 7 | \( 1 - 0.735T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 17 | \( 1 - 5.96T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 5.31T + 23T^{2} \) |
| 29 | \( 1 + 8.12T + 29T^{2} \) |
| 31 | \( 1 + 1.67T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 59 | \( 1 + 4.28T + 59T^{2} \) |
| 61 | \( 1 + 2.91T + 61T^{2} \) |
| 67 | \( 1 + 0.596T + 67T^{2} \) |
| 71 | \( 1 - 2.30T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 9.10T + 83T^{2} \) |
| 89 | \( 1 - 8.77T + 89T^{2} \) |
| 97 | \( 1 + 6.86T + 97T^{2} \) |
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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
−8.264311817543293700092118026135, −7.70738194191150754913516085148, −7.32507016216675852961706205000, −6.32147117028371872348662408590, −5.36102469624808980829449377019, −4.48796299814743107151708971411, −3.72743283637119550623362156495, −3.19831522720164251665746945645, −1.99192125140785734051341695848, −0.789997458263512926483671216196,
0.789997458263512926483671216196, 1.99192125140785734051341695848, 3.19831522720164251665746945645, 3.72743283637119550623362156495, 4.48796299814743107151708971411, 5.36102469624808980829449377019, 6.32147117028371872348662408590, 7.32507016216675852961706205000, 7.70738194191150754913516085148, 8.264311817543293700092118026135
