L(s) = 1 | + 8.89i·5-s − 2.64·7-s − 17.7i·11-s + 2.58·13-s − 25.8i·17-s − 20·19-s − 17.7i·23-s − 54.1·25-s + 11.9i·29-s + 17.1·31-s − 23.5i·35-s + 38·37-s − 15.7i·41-s + 43.4·43-s + 16.9i·47-s + ⋯ |
L(s) = 1 | + 1.77i·5-s − 0.377·7-s − 1.61i·11-s + 0.198·13-s − 1.52i·17-s − 1.05·19-s − 0.773i·23-s − 2.16·25-s + 0.410i·29-s + 0.553·31-s − 0.672i·35-s + 1.02·37-s − 0.383i·41-s + 1.01·43-s + 0.361i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.251840045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.251840045\) |
\(L(2)\) |
|
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 | \( 1 + 2.64T \) |
good | 5 | \( 1 - 8.89iT - 25T^{2} \) |
| 11 | \( 1 + 17.7iT - 121T^{2} \) |
| 13 | \( 1 - 2.58T + 169T^{2} \) |
| 17 | \( 1 + 25.8iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 + 17.7iT - 529T^{2} \) |
| 29 | \( 1 - 11.9iT - 841T^{2} \) |
| 31 | \( 1 - 17.1T + 961T^{2} \) |
| 37 | \( 1 - 38T + 1.36e3T^{2} \) |
| 41 | \( 1 + 15.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 43.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 85.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.64iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 100.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 36.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 28.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 118.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 120. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 44.4T + 9.40e3T^{2} \) |
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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
−9.803216407945812222703658456974, −8.804798057826675508936540037230, −7.916620720304039050948420782617, −6.89269866603355289701073064359, −6.42172383766949962213545817133, −5.57498455519493265308382495654, −4.07280237136952159045087608046, −3.06942873010825798844561053630, −2.52926102221368154612528690047, −0.42377298120459355952320691734,
1.18694485422708067150520389246, 2.17487563449298323206194427155, 4.06701630130423286310231960545, 4.44724056407436977730484003191, 5.54163392163479254332408827124, 6.36569825769702670703154799430, 7.58845262250339665650889920159, 8.309996063449082100187038666952, 9.089996739357434188235889710284, 9.747426251572554843130087184748