L(s) = 1 | + 3-s − 4.29·5-s − 4.35·7-s + 9-s + 1.15·11-s − 4.29·15-s + 0.493·17-s + 1.78·19-s − 4.35·21-s − 3.38·23-s + 13.4·25-s + 27-s + 6.93·29-s − 2.22·31-s + 1.15·33-s + 18.7·35-s + 3.87·37-s + 6.31·41-s − 7.38·43-s − 4.29·45-s − 1.78·47-s + 11.9·49-s + 0.493·51-s − 2.51·53-s − 4.97·55-s + 1.78·57-s + 6.63·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.92·5-s − 1.64·7-s + 0.333·9-s + 0.349·11-s − 1.10·15-s + 0.119·17-s + 0.408·19-s − 0.950·21-s − 0.705·23-s + 2.69·25-s + 0.192·27-s + 1.28·29-s − 0.399·31-s + 0.201·33-s + 3.16·35-s + 0.637·37-s + 0.986·41-s − 1.12·43-s − 0.640·45-s − 0.259·47-s + 1.71·49-s + 0.0691·51-s − 0.345·53-s − 0.671·55-s + 0.235·57-s + 0.863·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 4.29T + 5T^{2} \) |
| 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 17 | \( 1 - 0.493T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 - 3.87T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 1.78T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 + 0.374T + 73T^{2} \) |
| 79 | \( 1 + 2.65T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 0.835T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−7.47136927361591224175130285300, −6.87572597211980862974148226081, −6.42164365111096139385678381982, −5.31585151948677342241543528801, −4.24631137043058507937110218722, −3.86440267209392315835173520436, −3.20125765058347096709258691940, −2.62934516526089275953648380200, −0.979109843023808221524109209279, 0,
0.979109843023808221524109209279, 2.62934516526089275953648380200, 3.20125765058347096709258691940, 3.86440267209392315835173520436, 4.24631137043058507937110218722, 5.31585151948677342241543528801, 6.42164365111096139385678381982, 6.87572597211980862974148226081, 7.47136927361591224175130285300