L(s) = 1 | + 3-s + 3.23·5-s + 2·7-s + 9-s + 3.23·11-s − 4.47·13-s + 3.23·15-s − 6.47·17-s − 1.23·19-s + 2·21-s + 23-s + 5.47·25-s + 27-s − 8.47·29-s + 1.52·31-s + 3.23·33-s + 6.47·35-s + 7.70·37-s − 4.47·39-s − 2·41-s + 5.23·43-s + 3.23·45-s − 8.94·47-s − 3·49-s − 6.47·51-s + 12.1·53-s + 10.4·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.44·5-s + 0.755·7-s + 0.333·9-s + 0.975·11-s − 1.24·13-s + 0.835·15-s − 1.56·17-s − 0.283·19-s + 0.436·21-s + 0.208·23-s + 1.09·25-s + 0.192·27-s − 1.57·29-s + 0.274·31-s + 0.563·33-s + 1.09·35-s + 1.26·37-s − 0.716·39-s − 0.312·41-s + 0.798·43-s + 0.482·45-s − 1.30·47-s − 0.428·49-s − 0.906·51-s + 1.67·53-s + 1.41·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.287417415\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.287417415\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−10.72760156944907736465037893082, −9.518744550689925093707440384934, −9.348375219323625272132623334489, −8.253323762704423409176422504909, −7.11643520235476368381027366379, −6.28276421229212921213981041104, −5.14058158415863562694794264910, −4.19771441225091706246030281738, −2.49573011718335948348383433247, −1.73809279950613777488635123756,
1.73809279950613777488635123756, 2.49573011718335948348383433247, 4.19771441225091706246030281738, 5.14058158415863562694794264910, 6.28276421229212921213981041104, 7.11643520235476368381027366379, 8.253323762704423409176422504909, 9.348375219323625272132623334489, 9.518744550689925093707440384934, 10.72760156944907736465037893082