| L(s) = 1 | + 2-s − 0.552·3-s + 4-s + 5-s − 0.552·6-s + 2.28·7-s + 8-s − 2.69·9-s + 10-s + 0.669·11-s − 0.552·12-s − 3.24·13-s + 2.28·14-s − 0.552·15-s + 16-s + 4.76·17-s − 2.69·18-s + 6.06·19-s + 20-s − 1.26·21-s + 0.669·22-s + 8.52·23-s − 0.552·24-s + 25-s − 3.24·26-s + 3.14·27-s + 2.28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.318·3-s + 0.5·4-s + 0.447·5-s − 0.225·6-s + 0.864·7-s + 0.353·8-s − 0.898·9-s + 0.316·10-s + 0.201·11-s − 0.159·12-s − 0.899·13-s + 0.611·14-s − 0.142·15-s + 0.250·16-s + 1.15·17-s − 0.635·18-s + 1.39·19-s + 0.223·20-s − 0.275·21-s + 0.142·22-s + 1.77·23-s − 0.112·24-s + 0.200·25-s − 0.635·26-s + 0.605·27-s + 0.432·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.502664507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.502664507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 + 0.552T + 3T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 - 0.669T + 11T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 - 6.06T + 19T^{2} \) |
| 23 | \( 1 - 8.52T + 23T^{2} \) |
| 29 | \( 1 + 7.13T + 29T^{2} \) |
| 31 | \( 1 + 1.49T + 31T^{2} \) |
| 37 | \( 1 + 4.44T + 37T^{2} \) |
| 41 | \( 1 - 0.00655T + 41T^{2} \) |
| 43 | \( 1 - 8.43T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 1.51T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 1.68T + 61T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 3.43T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 5.13T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.85070400538262931425872830363, −7.33541468163893453734310638970, −6.60155972239226361780540323363, −5.49450143301636752524055461267, −5.37490239081621227937309724322, −4.76770356144723784140649061141, −3.52408049179241368096873672168, −2.94041335501465337294486288036, −1.93455166258233640588412070810, −0.933435974878968405938336412698,
0.933435974878968405938336412698, 1.93455166258233640588412070810, 2.94041335501465337294486288036, 3.52408049179241368096873672168, 4.76770356144723784140649061141, 5.37490239081621227937309724322, 5.49450143301636752524055461267, 6.60155972239226361780540323363, 7.33541468163893453734310638970, 7.85070400538262931425872830363
