| L(s) = 1 | + 1.57·3-s − 11.2·5-s − 14.3·7-s − 24.5·9-s − 42.8·11-s + 70.4·13-s − 17.7·15-s − 36.6·19-s − 22.5·21-s + 182.·23-s + 1.19·25-s − 81.2·27-s + 264.·29-s + 341.·31-s − 67.5·33-s + 160.·35-s + 65.5·37-s + 111.·39-s + 15.1·41-s + 149.·43-s + 275.·45-s − 4.93·47-s − 138.·49-s − 658.·53-s + 481.·55-s − 57.7·57-s + 105.·59-s + ⋯ |
| L(s) = 1 | + 0.303·3-s − 1.00·5-s − 0.772·7-s − 0.907·9-s − 1.17·11-s + 1.50·13-s − 0.304·15-s − 0.442·19-s − 0.234·21-s + 1.65·23-s + 0.00953·25-s − 0.578·27-s + 1.69·29-s + 1.98·31-s − 0.356·33-s + 0.776·35-s + 0.291·37-s + 0.456·39-s + 0.0576·41-s + 0.529·43-s + 0.912·45-s − 0.0153·47-s − 0.403·49-s − 1.70·53-s + 1.18·55-s − 0.134·57-s + 0.232·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 1.57T + 27T^{2} \) |
| 5 | \( 1 + 11.2T + 125T^{2} \) |
| 7 | \( 1 + 14.3T + 343T^{2} \) |
| 11 | \( 1 + 42.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 36.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 264.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 341.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 65.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 15.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 149.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 4.93T + 1.03e5T^{2} \) |
| 53 | \( 1 + 658.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 105.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 588.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 952.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 673.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 525.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 369.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 234.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 134.T + 9.12e5T^{2} \) |
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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
−8.357412257762245192722766819198, −7.71768085767230211891751053795, −6.63311826656791702639334952989, −6.07287799256014804438834206622, −5.00687241441531193582462211500, −4.12312843436806277391181360204, −3.05559051294090588528333619904, −2.81336246373006649385757560978, −1.00241807553019362335063175014, 0,
1.00241807553019362335063175014, 2.81336246373006649385757560978, 3.05559051294090588528333619904, 4.12312843436806277391181360204, 5.00687241441531193582462211500, 6.07287799256014804438834206622, 6.63311826656791702639334952989, 7.71768085767230211891751053795, 8.357412257762245192722766819198
