L(s) = 1 | − 1.73i·2-s − 0.999·4-s + i·7-s − 1.73i·8-s − 3.46·11-s − 2i·13-s + 1.73·14-s − 5·16-s − 3.46i·17-s + 4·19-s + 5.99i·22-s − 3.46i·23-s − 3.46·26-s − 0.999i·28-s − 4·31-s + 5.19i·32-s + ⋯ |
L(s) = 1 | − 1.22i·2-s − 0.499·4-s + 0.377i·7-s − 0.612i·8-s − 1.04·11-s − 0.554i·13-s + 0.462·14-s − 1.25·16-s − 0.840i·17-s + 0.917·19-s + 1.27i·22-s − 0.722i·23-s − 0.679·26-s − 0.188i·28-s − 0.718·31-s + 0.918i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.007604045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.007604045\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 6.92iT - 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.274152962676942347699771514842, −8.302657469105259898114240101090, −7.45927792318886330116166929473, −6.57118058147694727049515209640, −5.40876932501466487149725686108, −4.73212011506240788125907370076, −3.38179826786474273101642017891, −2.83370574803014797377813799926, −1.80952599511106560958530977976, −0.36662766755524675137682588130,
1.73794105781573257134743988567, 3.06047738782262066843252580806, 4.26624174581932617082446769737, 5.25612211714507900978673152750, 5.81767197148518574693924923274, 6.79569653954925309206715253795, 7.45155933382737355276126551737, 8.023834047903189408234689359085, 8.845681651361727110563730784040, 9.706109928992134410203835906099