| L(s) = 1 | − 3.67·3-s − 7.12·5-s − 0.511·7-s − 13.4·9-s + 17.5·11-s + 52.1·13-s + 26.1·15-s + 68.1·17-s − 19·19-s + 1.87·21-s + 48.8·23-s − 74.1·25-s + 148.·27-s − 87.1·29-s + 53.2·31-s − 64.5·33-s + 3.64·35-s − 199.·37-s − 191.·39-s − 372.·41-s + 416.·43-s + 96.2·45-s − 221.·47-s − 342.·49-s − 250.·51-s + 107.·53-s − 125.·55-s + ⋯ |
| L(s) = 1 | − 0.707·3-s − 0.637·5-s − 0.0276·7-s − 0.499·9-s + 0.481·11-s + 1.11·13-s + 0.450·15-s + 0.972·17-s − 0.229·19-s + 0.0195·21-s + 0.442·23-s − 0.593·25-s + 1.06·27-s − 0.558·29-s + 0.308·31-s − 0.340·33-s + 0.0176·35-s − 0.887·37-s − 0.786·39-s − 1.41·41-s + 1.47·43-s + 0.318·45-s − 0.687·47-s − 0.999·49-s − 0.687·51-s + 0.279·53-s − 0.307·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 3.67T + 27T^{2} \) |
| 5 | \( 1 + 7.12T + 125T^{2} \) |
| 7 | \( 1 + 0.511T + 343T^{2} \) |
| 11 | \( 1 - 17.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 48.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 87.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 53.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 199.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 372.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 416.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 107.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 155.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 229.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 580.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 38.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 803.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 641.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 19.9T + 9.12e5T^{2} \) |
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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
−9.933584570872444950565616672066, −8.799815546546678730920579068255, −8.122968443764359282410911735688, −7.02158356917907482922481641931, −6.08835354685652503481126197117, −5.32455961106673141695972062395, −4.07509333081071978830525758415, −3.15722967927310516109032309352, −1.33533005742931618238472936527, 0,
1.33533005742931618238472936527, 3.15722967927310516109032309352, 4.07509333081071978830525758415, 5.32455961106673141695972062395, 6.08835354685652503481126197117, 7.02158356917907482922481641931, 8.122968443764359282410911735688, 8.799815546546678730920579068255, 9.933584570872444950565616672066
