L(s) = 1 | + (−0.5 − 0.866i)3-s + (1.5 + 0.866i)7-s + 13-s + (0.5 − 0.866i)17-s − 1.73i·21-s + (0.5 + 0.866i)23-s − 25-s − 27-s + 1.73i·31-s + (−0.5 − 0.866i)39-s + (1 + 1.73i)49-s − 0.999·51-s + 53-s + (0.499 − 0.866i)69-s + (0.5 + 0.866i)75-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (1.5 + 0.866i)7-s + 13-s + (0.5 − 0.866i)17-s − 1.73i·21-s + (0.5 + 0.866i)23-s − 25-s − 27-s + 1.73i·31-s + (−0.5 − 0.866i)39-s + (1 + 1.73i)49-s − 0.999·51-s + 53-s + (0.499 − 0.866i)69-s + (0.5 + 0.866i)75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.392590064\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.392590064\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−8.624726037632819273908866235694, −7.83808324027748331147129865930, −7.28670472666953837477336870987, −6.44941422520343902276979060777, −5.55859966725343036712703497700, −5.24129221673672671022157121573, −4.12561112265413713083337269166, −3.02693619140794236647783355922, −1.80469297793667339808263610921, −1.22069847964624448554316543528,
1.15465668295810960971829390467, 2.18838843649263246156672064094, 3.87409583793628255342601356442, 4.07475688454418425507544440438, 4.96649468017496218892515731289, 5.63783045366243227147870847505, 6.43781592945529871974507883729, 7.60964343570548200382126728716, 7.965202154151111279028591665667, 8.748119077960381425914505505143