L(s) = 1 | + (−4.5 − 2.59i)7-s + (−2.5 − 2.59i)13-s + (3 + 1.73i)19-s + 5·25-s + 8.66i·31-s + (−6 + 3.46i)37-s + (−6.5 + 11.2i)43-s + (10 + 17.3i)49-s + (−6.5 + 11.2i)61-s + (10.5 − 6.06i)67-s − 1.73i·73-s − 13·79-s + (4.5 + 18.1i)91-s + (−16.5 − 9.52i)97-s + 13·103-s + ⋯ |
L(s) = 1 | + (−1.70 − 0.981i)7-s + (−0.693 − 0.720i)13-s + (0.688 + 0.397i)19-s + 25-s + 1.55i·31-s + (−0.986 + 0.569i)37-s + (−0.991 + 1.71i)43-s + (1.42 + 2.47i)49-s + (−0.832 + 1.44i)61-s + (1.28 − 0.740i)67-s − 0.202i·73-s − 1.46·79-s + (0.471 + 1.90i)91-s + (−1.67 − 0.967i)97-s + 1.28·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5254547477\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5254547477\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (2.5 + 2.59i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + (4.5 + 2.59i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 + (6 - 3.46i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.5 - 11.2i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.5 + 6.06i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (16.5 + 9.52i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.625404835215112473600895791159, −8.761757896854679554068394627585, −7.74533690991497304844805343422, −6.99117279212361253848842526326, −6.48911821246867901283856990383, −5.44932253342109961711376108924, −4.52000787618939290002677192193, −3.32246481759121778041600236476, −2.98776449439593926433129434646, −1.14214071696002958645883226899,
0.20903242480536027883324706855, 2.14947045124176899687830411569, 2.97947230813279821631773916377, 3.85673828228338419254397576347, 5.09010904166559153909563342951, 5.80869676095609006725140305897, 6.73439692079206936648473774906, 7.14951754536012708732776646912, 8.415828913659778977367154055652, 9.191828061747497082956040237728