L(s) = 1 | + (−0.618 + 1.90i)2-s + (0.809 − 0.587i)3-s + (−1.61 − 1.17i)4-s + (1.23 + 3.80i)5-s + (0.618 + 1.90i)6-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s − 7.99·10-s − 1.99·12-s + (−0.618 + 1.90i)13-s + (1.61 − 1.17i)14-s + (3.23 + 2.35i)15-s + (−1.23 − 3.80i)16-s + (1.23 + 3.80i)17-s + (1.61 + 1.17i)18-s + (2.42 − 1.76i)19-s + ⋯ |
L(s) = 1 | + (−0.437 + 1.34i)2-s + (0.467 − 0.339i)3-s + (−0.809 − 0.587i)4-s + (0.552 + 1.70i)5-s + (0.252 + 0.776i)6-s + (−0.305 − 0.222i)7-s + (0.103 − 0.317i)9-s − 2.52·10-s − 0.577·12-s + (−0.171 + 0.527i)13-s + (0.432 − 0.314i)14-s + (0.835 + 0.607i)15-s + (−0.309 − 0.951i)16-s + (0.299 + 0.922i)17-s + (0.381 + 0.277i)18-s + (0.556 − 0.404i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.202972 + 1.19625i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.202972 + 1.19625i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.618 - 1.90i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (-1.23 - 3.80i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (0.618 - 1.90i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.23 - 3.80i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.42 + 1.76i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + (4.85 + 3.52i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.54 - 4.75i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.42 + 1.76i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.61 + 1.17i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (1.61 - 1.17i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.85 + 5.70i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.09 - 5.87i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.927 - 2.85i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + T + 67T^{2} \) |
| 71 | \( 1 + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.89 - 6.46i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.39 + 10.4i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.85 - 5.70i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + (-1.54 + 4.75i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.70383487021972640585941986070, −10.67876981491949699471841163687, −9.752659191974828600016036016967, −8.950030244055664503676547827265, −7.71412500872165265114027978382, −7.09189010786535036187313322949, −6.47256653652740055567355236796, −5.58438627778757886434624205485, −3.58678988631292486509771966623, −2.36676814935249859874227146197,
0.938142977752789380568467966929, 2.28608464492803981083177642486, 3.56149018517137076201341915309, 4.83959908000021541539292423472, 5.80337222659154988070855351734, 7.70442642977038168650644535156, 8.796865143024675525707178496802, 9.396135499062318065773605361694, 9.819989387082137618049523400901, 10.95151528676278438983620826221