| L(s) = 1 | + 0.0996·2-s + 0.331·3-s − 1.99·4-s + 1.61·5-s + 0.0330·6-s + 7-s − 0.397·8-s − 2.89·9-s + 0.161·10-s + 5.37·11-s − 0.658·12-s − 4.14·13-s + 0.0996·14-s + 0.536·15-s + 3.94·16-s − 5.48·17-s − 0.288·18-s + 4.84·19-s − 3.22·20-s + 0.331·21-s + 0.536·22-s − 0.131·24-s − 2.37·25-s − 0.413·26-s − 1.95·27-s − 1.99·28-s − 7.39·29-s + ⋯ |
| L(s) = 1 | + 0.0704·2-s + 0.191·3-s − 0.995·4-s + 0.724·5-s + 0.0134·6-s + 0.377·7-s − 0.140·8-s − 0.963·9-s + 0.0510·10-s + 1.62·11-s − 0.190·12-s − 1.15·13-s + 0.0266·14-s + 0.138·15-s + 0.985·16-s − 1.33·17-s − 0.0679·18-s + 1.11·19-s − 0.720·20-s + 0.0722·21-s + 0.114·22-s − 0.0268·24-s − 0.475·25-s − 0.0811·26-s − 0.375·27-s − 0.376·28-s − 1.37·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.0996T + 2T^{2} \) |
| 3 | \( 1 - 0.331T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 13 | \( 1 + 4.14T + 13T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 29 | \( 1 + 7.39T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 - 3.48T + 41T^{2} \) |
| 43 | \( 1 + 7.51T + 43T^{2} \) |
| 47 | \( 1 + 9.08T + 47T^{2} \) |
| 53 | \( 1 - 8.08T + 53T^{2} \) |
| 59 | \( 1 - 0.680T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 8.07T + 73T^{2} \) |
| 79 | \( 1 + 7.78T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 8.81T + 97T^{2} \) |
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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
−8.351613670621288610053850518651, −7.46693693755592051742642358974, −6.63348634941995563721553399350, −5.69775410543845880150567962768, −5.23523078844166760469559983198, −4.28506023288546919027328417924, −3.59726389361658515911039732672, −2.46202473829271967051285152548, −1.48634885798574666976643141617, 0,
1.48634885798574666976643141617, 2.46202473829271967051285152548, 3.59726389361658515911039732672, 4.28506023288546919027328417924, 5.23523078844166760469559983198, 5.69775410543845880150567962768, 6.63348634941995563721553399350, 7.46693693755592051742642358974, 8.351613670621288610053850518651
