L(s) = 1 | + (1.16 − 1.16i)5-s − 3.74·7-s + (−1.64 − 1.64i)11-s + (−0.645 + 0.645i)13-s + 6.57i·17-s + (1.41 + 1.41i)19-s − 6i·23-s + 2.29i·25-s + (5.40 + 5.40i)29-s + 0.913i·31-s + (−4.35 + 4.35i)35-s + (1 + i)37-s + 6.57·41-s + (3.74 − 3.74i)43-s + 6·47-s + ⋯ |
L(s) = 1 | + (0.520 − 0.520i)5-s − 1.41·7-s + (−0.496 − 0.496i)11-s + (−0.179 + 0.179i)13-s + 1.59i·17-s + (0.324 + 0.324i)19-s − 1.25i·23-s + 0.458i·25-s + (1.00 + 1.00i)29-s + 0.164i·31-s + (−0.736 + 0.736i)35-s + (0.164 + 0.164i)37-s + 1.02·41-s + (0.570 − 0.570i)43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.385508708\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.385508708\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.16 + 1.16i)T - 5iT^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 + (1.64 + 1.64i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.645 - 0.645i)T - 13iT^{2} \) |
| 17 | \( 1 - 6.57iT - 17T^{2} \) |
| 19 | \( 1 + (-1.41 - 1.41i)T + 19iT^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + (-5.40 - 5.40i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.913iT - 31T^{2} \) |
| 37 | \( 1 + (-1 - i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.57T + 41T^{2} \) |
| 43 | \( 1 + (-3.74 + 3.74i)T - 43iT^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (7.73 - 7.73i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9.29 - 9.29i)T + 59iT^{2} \) |
| 61 | \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \) |
| 67 | \( 1 + (-10.8 - 10.8i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.70iT - 71T^{2} \) |
| 73 | \( 1 + 12.5iT - 73T^{2} \) |
| 79 | \( 1 + 14.0iT - 79T^{2} \) |
| 83 | \( 1 + (10.9 - 10.9i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.412T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−8.965441347663559537043473548755, −8.558320858017560174000910854055, −7.52127102915781601762044254842, −6.54067133131812786143153441200, −6.03192812555595269505354678348, −5.26500865907917670186138826027, −4.16113425053943199073467155695, −3.28092470614726995870346241366, −2.34337673724101803657572013475, −0.958599761603874227303465315046,
0.58458808041660940700985721623, 2.46321727503462380319352957986, 2.85649823022868980166314881289, 3.97398386724047126428295919232, 5.10123264577764550380017309387, 5.85212196859519090865624584572, 6.70919142432848075306209266111, 7.20080433889967376136539695063, 8.066451600965663459844659636523, 9.256692307406143169461581431112