L(s) = 1 | + (−1.22 − 1.22i)3-s + (−3.67 + 3.67i)7-s + 2.99i·9-s + (−1.22 − 1.22i)13-s − 7i·19-s + 9·21-s + (3.67 − 3.67i)27-s + 11·31-s + (4.89 − 4.89i)37-s + 2.99i·39-s + (1.22 + 1.22i)43-s − 20i·49-s + (−8.57 + 8.57i)57-s − 61-s + (−11.0 − 11.0i)63-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s + (−1.38 + 1.38i)7-s + 0.999i·9-s + (−0.339 − 0.339i)13-s − 1.60i·19-s + 1.96·21-s + (0.707 − 0.707i)27-s + 1.97·31-s + (0.805 − 0.805i)37-s + 0.480i·39-s + (0.186 + 0.186i)43-s − 2.85i·49-s + (−1.13 + 1.13i)57-s − 0.128·61-s + (−1.38 − 1.38i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7910190063\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7910190063\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3.67 - 3.67i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (1.22 + 1.22i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 11T + 31T^{2} \) |
| 37 | \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-1.22 - 1.22i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + (-8.57 + 8.57i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (9.79 + 9.79i)T + 73iT^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-3.67 + 3.67i)T - 97iT^{2} \) |
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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
−9.546572881653600411604133466526, −8.844485665074644456089298205751, −7.88540472422870268425624347373, −6.83731682203099200377387225402, −6.30234720244264038928220051807, −5.55004746568696187073607647491, −4.66070161048579647595081844975, −2.98666061868708644707400269449, −2.35031734485152074039002646127, −0.48230464576594586914570394977,
0.949317840475400380255940887836, 3.03586657001468690597935073141, 3.93053643832338263080004736896, 4.54908460597550949221496140204, 5.86250548661354191679816655373, 6.48149732173987194076299816332, 7.22310816672236021576593933063, 8.293263400848682666183933861284, 9.527914530561104566105047828341, 10.03113149576747701767374655805