| L(s) = 1 | + 2-s − 0.688·3-s + 4-s + 3.20·5-s − 0.688·6-s − 0.886·7-s + 8-s − 2.52·9-s + 3.20·10-s + 11-s − 0.688·12-s − 4.69·13-s − 0.886·14-s − 2.20·15-s + 16-s − 3.36·17-s − 2.52·18-s + 2.77·19-s + 3.20·20-s + 0.610·21-s + 22-s − 9.05·23-s − 0.688·24-s + 5.29·25-s − 4.69·26-s + 3.80·27-s − 0.886·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.397·3-s + 0.5·4-s + 1.43·5-s − 0.281·6-s − 0.335·7-s + 0.353·8-s − 0.841·9-s + 1.01·10-s + 0.301·11-s − 0.198·12-s − 1.30·13-s − 0.237·14-s − 0.570·15-s + 0.250·16-s − 0.816·17-s − 0.595·18-s + 0.635·19-s + 0.717·20-s + 0.133·21-s + 0.213·22-s − 1.88·23-s − 0.140·24-s + 1.05·25-s − 0.921·26-s + 0.732·27-s − 0.167·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 0.688T + 3T^{2} \) |
| 5 | \( 1 - 3.20T + 5T^{2} \) |
| 7 | \( 1 + 0.886T + 7T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 23 | \( 1 + 9.05T + 23T^{2} \) |
| 29 | \( 1 + 5.35T + 29T^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 - 1.56T + 41T^{2} \) |
| 43 | \( 1 + 5.19T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 + 5.41T + 53T^{2} \) |
| 59 | \( 1 + 5.49T + 59T^{2} \) |
| 61 | \( 1 - 2.95T + 61T^{2} \) |
| 67 | \( 1 + 6.89T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 8.58T + 73T^{2} \) |
| 79 | \( 1 + 2.43T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 8.96T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78853118044503173552383020060, −7.09303817350736171724246540614, −6.16206069602164782328459389715, −5.86366914033855513353101834635, −5.22280506558699729221568541488, −4.39302899906668080879586530073, −3.31536495059769038149223648816, −2.37559831065828186784863551237, −1.80800894149388536422592047430, 0,
1.80800894149388536422592047430, 2.37559831065828186784863551237, 3.31536495059769038149223648816, 4.39302899906668080879586530073, 5.22280506558699729221568541488, 5.86366914033855513353101834635, 6.16206069602164782328459389715, 7.09303817350736171724246540614, 7.78853118044503173552383020060
