| L(s) = 1 | − 2-s + 4-s + (0.426 + 2.19i)5-s − 8-s + (−0.426 − 2.19i)10-s + 1.28i·11-s + 6.14·13-s + 16-s − 6.52i·17-s + 6.03i·19-s + (0.426 + 2.19i)20-s − 1.28i·22-s − 2.87·23-s + (−4.63 + 1.87i)25-s − 6.14·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.190 + 0.981i)5-s − 0.353·8-s + (−0.134 − 0.694i)10-s + 0.386i·11-s + 1.70·13-s + 0.250·16-s − 1.58i·17-s + 1.38i·19-s + (0.0953 + 0.490i)20-s − 0.273i·22-s − 0.600·23-s + (−0.927 + 0.374i)25-s − 1.20·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.218969495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218969495\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.426 - 2.19i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 1.28iT - 11T^{2} \) |
| 13 | \( 1 - 6.14T + 13T^{2} \) |
| 17 | \( 1 + 6.52iT - 17T^{2} \) |
| 19 | \( 1 - 6.03iT - 19T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 - 1.35iT - 29T^{2} \) |
| 31 | \( 1 - 8.65iT - 31T^{2} \) |
| 37 | \( 1 - 9.53iT - 37T^{2} \) |
| 41 | \( 1 + 8.71T + 41T^{2} \) |
| 43 | \( 1 + 5.35iT - 43T^{2} \) |
| 47 | \( 1 + 0.806iT - 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 59 | \( 1 + 1.59T + 59T^{2} \) |
| 61 | \( 1 + 6.35iT - 61T^{2} \) |
| 67 | \( 1 + 5.39iT - 67T^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 3.74iT - 83T^{2} \) |
| 89 | \( 1 - 3.63T + 89T^{2} \) |
| 97 | \( 1 + 8.76T + 97T^{2} \) |
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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
−8.472896814057612783145080496549, −8.007745842664651067570148299748, −7.00840176203626130351259913927, −6.66444633326545359620653213654, −5.86458345470106024825422604631, −5.03735308238506809820019229853, −3.69240271485531437401005913900, −3.22520620934763206458082786284, −2.11543053779627291614667085063, −1.21972795199626931034481142081,
0.46625145767607682321401642447, 1.42669566128243200415796661143, 2.29711493049257491090039551978, 3.65428278402981246120271720968, 4.19036315218416803936437976452, 5.37225954396302670805684194084, 6.05727674676168214417389071927, 6.52223946870003636763994403372, 7.75043837680216099060165836884, 8.237420235436202974583119470648
