L(s) = 1 | − 2.21i·5-s + i·7-s + 0.614·11-s − 5.56·13-s + 0.585i·17-s − 3.89i·19-s + 0.850·23-s + 0.0963·25-s + 4.74i·29-s + 3.07i·31-s + 2.21·35-s + 2.17·37-s − 6.07i·41-s − 9.21i·43-s + 3.41·47-s + ⋯ |
L(s) = 1 | − 0.990i·5-s + 0.377i·7-s + 0.185·11-s − 1.54·13-s + 0.142i·17-s − 0.894i·19-s + 0.177·23-s + 0.0192·25-s + 0.880i·29-s + 0.551i·31-s + 0.374·35-s + 0.357·37-s − 0.949i·41-s − 1.40i·43-s + 0.498·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1802044023\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1802044023\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 2.21iT - 5T^{2} \) |
| 11 | \( 1 - 0.614T + 11T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 17 | \( 1 - 0.585iT - 17T^{2} \) |
| 19 | \( 1 + 3.89iT - 19T^{2} \) |
| 23 | \( 1 - 0.850T + 23T^{2} \) |
| 29 | \( 1 - 4.74iT - 29T^{2} \) |
| 31 | \( 1 - 3.07iT - 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 + 6.07iT - 41T^{2} \) |
| 43 | \( 1 + 9.21iT - 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 - 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 4.83iT - 67T^{2} \) |
| 71 | \( 1 - 5.33T + 71T^{2} \) |
| 73 | \( 1 + 9.28T + 73T^{2} \) |
| 79 | \( 1 - 1.29iT - 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−8.470697791636649508843040085253, −7.55569672737189024083878294688, −7.06700739464062265850501530398, −6.17854034610057696570148782150, −5.14764288468677140425724604209, −4.98910244715915745001338336407, −4.10348745094942347156076789769, −3.00508279960514779968296908214, −2.21407394102098134448469724813, −1.14398684132876561380804447281,
0.04667426789876708475996127399, 1.53969361647884991701933662989, 2.63405569111159572147835280757, 3.12639352942696345035140030263, 4.22830980979654164006182449302, 4.76354443235916262603031847986, 5.86514481279910528626428603937, 6.38521557257854512805747326656, 7.28240127978392551947883795845, 7.56533314793010342402243215766